The propagation of poroelastic waves in a soil specimen is dependent on the physical and mechanical properties of the soil. In the geotechnical practice, such properties are obtained through in-situ geotechnical testings or element soil testings in the laboratory. These methods require almost advanced equipment and both testing and sample preparation may be expensive and time-consuming. This paper aims to present an algorithm for a laboratory-scale ultrasonic non-destructive testing to determine the physical and mechanical properties of saturated soil samples based on the distribution of stress waves. The ultrasonic setup, in comparison to most conventional soil lab testing equipment, is low-cost and non-invasive such that it reduces the soil disturbance. For this purpose, a poro-elastodynamic forward solver and differential evolution global optimization algorithm were applied to characterize the porosity, density, and other mechanical properties for a soil column. The forward solver was developed based on a semi-analytical solution which does not require intensive computational efforts encountered in standard numerical techniques such as the finite element method. It was concluded that the proposed high-frequency ultrasonic technique characterizes effectively the saturated soil samples based on the output stress wave measured by the receiver. This development makes geotechnical investigations time-efficient and cost-effective, and as such more suited to applications in remote areas. Characterizing foundation soils is the first step in design and construction of civil infrastructure. The measurement of physical and mechanical properties of soils  requires intensive in-situ or laboratory tests, which can be time consuming and costly. Soil samples, especially from projects in remote areas, are required to be transported to a geotechnical laboratory for various tests. This can cause the disturbance of soil samples, and laboratory tests on disturbed samples may lead to erroneous conclusions.1. The bender element method was developed in the 1980s and its simplicity is widely recognized: one transducer is placed at one end of a soil specimen for the generation of stress waves; one receiver is placed at the other end to record the induced stress waves. Various interpretation methods have been proposed in the past. The shear wave velocity can be calculated from the time difference between the input and output waves by assuming the absence of reflected or refracted waves2. However, it is well known that the identification of the arrival time of the output wave is subjective1. Other signal processing techniques such as the cross-correlation of the input and output stress waves3 and the second arrival of the output wave4 are based on the peak values of the stress wave for the estimation of the shear wave velocity. Some other methods  are used for estimating the relation between the phase angle and shear wave velocity in the frequency domain.The laboratory methods for measuring the shear wave velocity of soil samples include the resonant column test, bender element test among others. However, there is no established standard developed for the interpretation of the dynamic test results9 through the finite difference, finite element, and discrete element methods to interpret the output stress waves. The elastodynamic algorithm assumes that the domain is composed of solid materials. Under a dynamic load, the generated P waves and S waves penetrate into different layers of a soil and the reflected waves received at the receiver can be used to determine the soil strata. However, the estimation of the shear wave velocity is still based on the simple signal processing techniques. In addition, in elastodynamic algorithms, the effect of porous structure of soil layers and pore water in dynamic responses of geomaterials is neglected. In fact, the wave propagation in porous soil layers can be better represented by using dynamic poroelastic models instead of elastodynamic models, especially in fully saturated soils in which the pore water can significantly attenuate the stress waves, and in high frequency regimes. The dynamic poroelastic models consider the coupling effect between the pore water and solid skeleton, which induces three types of waves . Under an impact load, those three waves travel at different speeds, which are captured by the receiver placed at the end of the soil specimen in an ultrasonic setup.The elastodynamic theory has been also used by several researchers11 has been solved using various analytical and numerical methods. A direct boundary element approach for solving three-dimensional problems of dynamic poroelasticity in the time domain was developed by12. Such a technique was based on an integral equation formulation in terms of solid displacements and fluid stress. The 2D and 3D fundamental solutions of dynamic poroelasticity was further developed by16. The solutions were obtained in both time and Laplace transform domain, and can be recovered to elastodynamics and steady-state poroelasticity. In layered saturated media, similar approaches have been reported by18. Other than the boundary element method, the finite element method has also been applied by19. The finite difference method is also used to simulate the wave propagation in heterogeneous poroelastic media by20.The problem of dynamic poroelasticityIn a conventional geotechnical apparatus used to determine the dynamic properties of a soil specimen, the focus is mainly on the estimation of shear wave velocity and the interpretation method is mostly based on the time interval difference between the input and output stress waves. To the best of our knowledge, there is currently no laboratory-scale ultrasonic setup which is able to determine a range of physical and mechanical properties of a soil sample. Furthermore, the development of cheaper, faster and portable means of soil characterization may significantly lower the cost of overall soil testing, and make better assessments possible in sensitive locations.This paper aims to present an ultrasonic-based poroelastodynamic algorithm, which can be used in an ultrasonic setup to determine a range of physical and mechanical properties of a soil sample such as shear wave velocity, compression wave velocity, density and porosity. Such a setup can also be used for in-situ geotechnical investigation on extracted soil samples. In this algorithm, the poro-elastodynamic forward solver for the characterization of soil samples in high frequency regimes is developed using the spectral element method. Such a meshless semi-analytical technique reduces significantly the computational efforts by avoiding unnecessary calculations for the entire domain. Instead, only the response at the receiver location is calculated, which will then be used during the optimization process. A robust global optimization algorithm is then applied to predict the soil properties given the stress signal measured by the receiver.A general schematic of the problem is illustrated in Fig. u is the displacement vector of the solid skeleton; w is the fluid displacement relative to the solid skeleton; λ and μ are the Lamé constants; α is the Biot coefficient; p is the pore-water pressure; M is 1/Kf is the bulk modulus of the fluid; Ks is the bulk modulus of the solid skeleton and ϕ is the porosity. λc = λ + α2M; m = ρfβ∕ϕ in which β is the tortuosity which is used to describe the diffusion properties in porous media, and ρf is the density of pore-water, taken as 1000 kg∕m3. The drag-force damping coefficient b is calculated as21: η is the fluid dynamic viscosity and κ is the permeability coefficient; F is the viscous correction factor22: Ms is taken as 1; ω is the angular frequency.By assuming the infinitesimal deformation of solid skeleton, the dynamic poroelastic governing equations are written as follows:eiωt− in which ω is the frequency and t denotes time variable. The governing equations in Laplace domain can be obtained by replacing ω with  − is where s is the Laplace variable.The governing equations can be written in frequency domain through the Fourier transform by performing convolution with ϕ) and a vector potential 5b)) in aIt can be seen from Eq.  that \doP is the eigenvector matrix and D is the eigenvalue matrix of the K matrix: The K matrix can be rewritten using the Eigen decomposition method: P due to the existence of the term P−1. Introducing Eq. 5d). By rC is the coefficient to be determined from the boundary conditions and J1 is the Bessel function of the first kind of order one. For simplicity, the term ks.Under axisymmetric conditions, the solution for Eq.  in the cr, θ, z), considering the axisymmetric conditions θ direction that does not vanish. For simplicity, the vector potential θ direction is denoted as In the cylindrical coordinates , the displacement components Similarly, the matrix for effective stress components and porewater pressure in frequency domain is shown in Eq. N matrix is applied to matrix Gi matrix, in which i denotes the layer number.According to the Cauchy stress principle, the traction force  is applied to one end of the soil specimen. The surface is assumed to be permeable, which implies the porewater pressure at the surface is zero. Under such conditions, the displacements in the frequency domain can be written as: In the ultrasonic tests, a vertical impulse load f is firstly defined in time domain and can decomposed into two independent functions in terms of time variable fn(t) and radial variable fr(r): The impulse load fn(t) depends mainly on the type of impulse loads created by the signal generator. In this paper, a sinusoidal impulse function is used as the external load to simulate the applied load. The load with amplitude of one is mathematically described in Eq. (t(s) is time and f(Hz) is the frequency; H is the Heaviside step function.The mathematical expression for the function d in Eq. .26\documfr(r) is normally written using the Fourier-Bessel series: r0 is the radius of the contact area; km is the mode number; n is the total mode number; r∞ is the diameter of the soil specimen.Meanwhile, the function 23: The displacement obtained in Eq.  is in thcm at the center of the top end of the soil column. The displacement at the center (r = 0) in the other end is recorded and compared.The characterization of porosity has been a challenge because soil porosity can not be captured through traditional low-frequency tests. Such limitations can be explained by comparing the size of pore space and wavelength. A sensitivity analysis of the soil porosity is performed to verify such limitations. In this study, a soil column with a height and radius of 0.1 m is studied. The impulse load is applied to an area with a radius of 127. For example, high-plasticity clay (CH based on the Unified Soil Classification System (USCS)) has a Young’s modulus ranging from 0.35 to 32 MPa and porosity from 0.39 to 0.59; Silts and clays of low plasticity  have a typical value of Young’s modulus ranging from 1.5 to 60 MPa and porosity from 0.29 to 0.56; poorly graded sands (SP) normally have a Young’s modulus from 10 to 80 MPa and porosity from 0.23 to 0.43; The Young’s modulus of well-graded gravel (GW) is between 30-320 MPa and its porosity is from 0.21 to 0.32. The average dry density ranges from 1700 to 2300 kg∕m3. The average permeability varies from 5 × 10−10 (clay of high plasticity) to 0.4 m/s (sand and gravel). The typical values of Poisson’s ratio vary from 0.1 to 0.49 for clay and from 0.3 to 0.35 for silt.The typical values of Young’s modulus, porosity, density, permeability and Poisson’s ratio are well documented in the literatureIn this paper, two groups of soils are studied: the first group includes clay, silt, sand and loose gravel which generally have a relatively low Young’s modulus (lower than 100 MPa). The second group includes dense gravel which has a Young’s modulus equal or greater than 200 MPa.kg∕m3. The wavelength can be calculated using the algorithm shown in Appendix C. Several sensitivity analyses under three impulse loads with various predominant frequencies are performed. The impulse load distributions in time and frequency domains are shown in Fig. The effect of impulse load frequency and soil parameters on the dynamic soil response is studied in this section for the above-mentioned groups of soils. For the first group, the soil properties are taken as: Young’s modulus is 20 MPa; Poisson’s ratio is 0.35; dry density is 1800 μm as reported by28. Through the root search algorithm described in Appendix C, the wavelength under the load 3 is calculated around 1000–2000 μm, which is close to the poro-space scale of the studied soil. Therefore, the impulse load 3 is a good choice for the lab-scale characterization of soil specimens for group 1.As shown in Fig. Similarly, the sensitivity analyses are performed by considering different densities, Young’s modulus and Poisson’s ratios. The output displacement is shown in Fig. In the case of soil group 2, dense gravel whose Young’s modulus is up to 320 MPa, it is found that the load 3 (up to 5kHz) generates similar displacement outputs at different porosities , as shown in Fig. In this section, a case study is presented to show the process of saturated soil characterization. For this purpose, a synthetic data is firstly generated to simulate real measurements. For simplicity, the results are only presented for soil group 1. The nature of this inversion problem and inversion algorithm selection are discussed in detail in the following sections. At the end, the inversion results (soil parameters) are given based on the synthetic data and selected inversion algorithm.kg/m3 and porosity is taken as 0.3; The time interval is set to be 2 ms. Under the impulse load 3, as shown in Fig. A synthetic data (the displacement measured by a piezoelectric receiver) is firstly obtained using the following settings: Young’s modulus is 20 MPa; Poisson’s ratio is 0.35; density of solid skeleton is 1800 The response measured at the receiver location is summarised in Fig. The inversion algorithm takes the measured displacement at the receiver location . Therefore, the gradient based method is preferred in most cases, especially for convex optimization problems. However, such a method is highly likely to be affected by the local minimum since the gradient at any local minimum is zero. Thus, it is not favorable for non-convex problems.An analysis was performed to show the nature of the soil characterization optimization problem. It is important to determine whether such application belongs to convex or non-convex problem. Then the corresponding optimization algorithm can be selected based on the nature of the problem. The aim (cost) function is defined as the Euclidean norm between the synthetic and predicted data. The optimization space can be visualized by performing parameter sweep. For example, the optimization space for the porosity and Poisson’s ratio is shown in Fig. 29.It is shown in Fig. 30.A brief description of the differential evolution algorithm is given in Fig. kg∕m3; porosity is 0.3 and loss function is 0. It can be seen that the prediction of soil parameters based on the transmitted wave measured by the receiver  shown in Fig. The differential evolution algorithm successfully finds the global minimum, despite of the existence of multiple local minimum. The spatial distribution of soil parameters updates are shown in Figs. The predicted soil properties  are likely to be affected by the noise level of the measurement data, which could be introduced by the sensor measurement errors and ambient noise. In this uncertainty analysis, random white noise is added to measured displacement data with targeted signal-to-noise (SRN) ratio. For example, the noisy data with 10 and 20 dB of SRN is shown in Fig. In addition, the uncertainty can be introduced by the unknown coupling performance in the interface of piezoelectric sensors and soil specimens. The input electricity signal does not necessarily generate the desired input pressure. To account for such uncertainties, the magnitude of input load is assumed to be in normal distribution, as shown in Fig. 31 is used for the uncertainty analysis in this paper. The PCE technique, as a rigorous uncertainty quantification method, provides reliable numerical estimates of uncertain physical quantities. It was also reported that the PCE is much faster than Monte Carlo methods when the number of uncertainty parameters are lower than 2032. The 90% confident interval of the displacement at the receiver location is calculated through the PCE technique, shown in Fig. The generalized Polynomial Chaos Expansions (PCE) method developed byThen, based on the inversion analysis, the predicted soil properties in the 90% confidence interval are shown in Table In this paper, an ultrasonic-based characterization of soil specimens is developed for the instant measurement of soil properties including Young’s modulus and Poisson’s ratio (compression/shear wave velocity), density and porosity. The developed meshless semi-analytical algorithm reduces the computational effort significantly in comparison to standard numerical techniques such as the finite element method. In fact, the advantage of such a solution is that the dynamic response is evaluated at the receiver location only rather than the entire domain. The soil response in other locations is not measured in the real application and does not factor in soil characterization.It is concluded that high-frequency impulse loads (with predominant frequency of up to 5 kHz) is required to trigger the effect of porosity for soils with relatively low Young’s modulus . For stiffer materials, such as very dense gravels, an impulse load with predominant frequency of 0.5 MHz is required to characterize their porous nature. The characterization of soil properties has been proved as a highly non-convex optimization problem in this paper. The differential evolution algorithm, as a global optimization method, is found efficient and effective in finding the optimum soil properties, such that the difference between the predicted and measured stress waves is minimized. In conclusion, the developed method in interpreting dynamic response of saturated soil can be used for the immediate characterization of Young’s modulus, Poisson’s ratio, density and porosity for a given soil specimen.The components of the matrix G1 and G2 are matrix for the first and second layer, respectively.ϵ for the wavenumber sweep increment; n for the number of iterations; k0 for the initial wavenumber, k for the wavenumber at the current step; f(k) gives the determinant value of the stiffness matrix at wavenumber k; δ for the tolerance used to check if the determinant of the stiffness matrix is close to zero; f(k) and r is the root calculated from Brent function.The algorithm performs a sweep in a broad range of wavenumbers for a given frequency. A rough interval where roots exist needs to be found first and then the classic Brent’s method can be applied to accurately locates the roots. The following notations are used in the algorithm: The algorithm is shown as follows:


OUES was also positively correlated with Based on the wide range of problems to effectively perform cardiopulmonary testing in young children, this study strives to develop a new cardiopulmonary exercise test for children using a mobile testing device worn in a backpack in order to test children during their natural movement habits, namely, running outdoors. A standard cardiopulmonary exercise ramp test on a cycle ergometer was performed by a group of twenty 7–10-year-old children. The results were compared with a self-paced incremental running test performed using a mobile cardiopulmonary exercise measuring device in an outdoor park. The children were able to reach significantly higher values for most of the cardiopulmonary exercise variables during the outdoor test and higher. Whereas a plateau in  Cardiopulmonary exercise testing (CPET) is a commonly performed, non-invasive method to evaluate cardiac symptoms and assess functional capacity in children  and can Cardiorespiratory fitness (CRF) is the most important marker of health among the health-related physical fitness components in children and adolescents , 4. TherThis is a consequence of the fact that only few studies have reported undertaking cardiopulmonary exercise testing in children less than 8 years of age –11. TherTreadmill testing has the advantage of extensive clinical use and is a well-established mode of exercise testing in a pediatric setting . In addient}V˙O2 , 14, 15 ent}V˙O2 . Still, ent}V˙O2 . Classicent}V˙O2 , which iexercise . The widexercise . For chiexercise . Furtherexercise .As a consequence to the abovementioned limitations of TM and CE testing on preschool children several authors have therefore aimed at developing field-based tests to indirectly estimate the ge group , like thent}V˙O2 .With the evolvement of mobile cardiopulmonary exercise testing accurate measurements of Twenty healthy children aged between seven and ten years of age agreed to participate in this study. All study participants as well as their legal guardians gave informed consent to participate in the study and for their data to be used for scientific analysis. The study as well as the parent and age-adapted children consent forms were approved by the Ethics Committee of the University of Erlangen-Nuremberg, FRG (35_18B). All participants were Caucasian, non-obese, and healthy. None took medications. The subjects were recruited from local schools. There was no attempt to recruit subjects who were particularly active.Height and weight were measured using a stadiometer and electronic scale . Body composition was evaluated using a bioimpedance measuring device .The parents were asked about school transport habits . Especially cycling to and from school has proven to positively influence cardiorespiratory fitness in school children , 22, wheA small, low-dead-space respiratory valve (88 ml) with a pediatric mouthpiece and headgear was used. Gas exchange was measured continuously during each test using a breath-by-breath method and averaged over 15 s intervals. The criteria for completion of peak V˙O2peak . All chiV˙O2peak  was obse1 and VT2 were determined using the V-slope method proposed by Wasserman et al. [Ventilatory thresholds VTn et al. . OUES waanalysis .The cycle ergometry was chosen as it represents the standard exercise testing mode in Europe. All subjects underwent the cycle test first in order to record a valid 12-lead ECG at rest and during the test , which was not possible during the outdoor setting. All tests were undertaken in the morning. The Cyclus 2, an electronically braked cycle ergometer  with a standard children’s mountain bike for children aged 8–10 years  was used for the cycle test. Before each test the height of the saddle and the handlebars were adjusted to the height of each child. Subjects warmed up on the bike for a period of 5 min. After a short rest period (10 min) subjects were fitted with a heart rate (HR) monitor . The mask was fitted. The same mobile exercise equipment was used for the CE test and for the outdoor test . The children began by cycling at 10 W for a 2 min warm-up phase. Work rate was then continuously incremented in a linear ramp pattern with an increase of 10 W/min, a protocol comparable to one proposed by Cooper et al.  but withThe outdoor test was performed a minimum of two days and maximum of two weeks after the bicycle test in order to guarantee a sufficient recovery but also to keep a comparable fitness level and anthropometry at the time of the second test. All tests were again carried out in the morning at or around the same time that each child had been tested on the CE. The climatic conditions were comparable to the cycle tests but not identical, as humidity, wind speed and even temperature cannot be controlled in an outdoor setting. The children warmed up in the same way as for the CE and then fitted with the equipment as described before. The test was designed as a self- or researcher-paced incremental test rather than a ramp or true step test. Each child was instructed prior to the test. The first increment would consist of walking at a leisurely pace. After a period of 2 min the speed would be increased to an easy jog (2 min), then to running with some effort (2 min). This speed was described as a speed they could be able to maintain for a longer period but should feel tiring. Lastly they would have to try to run as fast as possible for as long as possible, so as if chasing another child who could take away their prize. An experienced researcher and running coach for children performed the whole test alongside each child controlling the speed and the time of each step using a watch equipped with a GPS sensor  in order to slow down or motivate the respective child if need be. All tests were accompanied by the same team. The speed was adapted to the capacities of each child by observation of as well as feedback from the child. A GPS sensor integrated in the mobile unit recorded the speed of the child at every moment of the test. Running was performed on a flat, wide and even trail over the whole distance of the test.Statistical analysis was performed using Microsoft Excel 2000® for data collection and SPSS 12.0® . All measured values are reported as means and standard deviations. The Kologomorov-Smirnov test was used to check for normal distribution. Homogeneity of variance was investigated using Levine’s F-test. For normally distributed variables differences between the two test protocols were assessed with paired t-tests, otherwise the Wilcoxon or the Whitney–Mann U tests were used. All tests were 2 tailed, a 5% probability level was considered significant.Pearson or Spearmen correlation coefficients were used to investigate univariate correlation between independent variables and We were able to test 20 children on the CE but only 19 outdoors as one girl caught a viral infection between the two tests. Out of the 20 children 12 were boys and 8 were girls. Their age, anthropometric data, as well as their sports participation and their school transport habits are represented in Table Both tests were well tolerated by the children. The main reason for stopping the CE test was weakness in the legs and most outdoor tests were ended when the speed during the last step could not further be increased and children reported being exhausted.During the CE 15 (75%) children reached the second ventilatory threshold (VT2), whereas during the outdoor test all children reached VT2 and a realistic  maximum  was reac2 pulse  similar to the standard heart rate monitors used during physical activity. Since our study has been conducted, Cortex Germany has integrated Custo Guard into their software so that a 3-channel ECG can now be recorded even during outdoor testing. So far this test has only been carried out in a group of healthy children. When implementing this kind of test in a patient group typical for pediatric cardiology like patients with Fontan circulation, or patients with cardiomyopathy, the test needs to be adapted to their cardiorespiratory fitness with regards to the achievable speeds in each step. More experience and more studies are therefore needed to implement this test in a patient group. However, when accompanying the children during the test, the visual and verbal feedback from the children to the medical supervisor allowed for good adaptation to the respective speeds achievable by each child. We believe that after acquiring some experience with this kind of outdoor test, conducting the same test in a group of patients with heart disease is feasible when special attention is paid to the visual and verbal feedback provided by each patient. Of course when testing patients, a certain backup of medical equipment is crucial. During our study we had an AED  and backpack with emergency equipment on site. However, patients need to be tested in close proximity to a care facility so that in a case of emergency all necessary equipment is readily available.As the children ran up to 12 km/h during the last step of the test, the medical supervisor needs to exhibit a certain degree of fitness in order to keep up with the children.Incremental exercise testing of children between 7 and 10 years of age in the park using a mobile cardiopulmonary exercise testing unit allows for accurate determination of


In this study we used meteorological parameters and predictive modelling interpreted by model explanation to develop stress metrics that indicate the presence of drought and heat stress at the specific environment. We started from the extreme temperature and precipitation indices, modified some of them and introduced additional drought indices relevant to the analysis. Based on maize’s sensitivity to stress, the growing season was divided into four stages. The features were calculated throughout the growing season and split in two groups, one for the drought and the other for heat stress. Generated meteorological features were combined with soil features and fed to random forest regression model for the yield prediction. Model explanation gave us the contribution of features to yield decrease, from which we estimated total amount of stress at the environments, which represents new environmental index. Using this index we ranked the environments according to the level of stress. More than 2400 hybrids were tested across the environments where they were grown and based on the yield stability they were marked as either tolerant or susceptible to heat, drought or combined heat and drought stress. Presented methodology and results were produced within the Syngenta Crop Challenge 2019. Zea mays L.) is one of the world’s most important crops. Every year, breeders create a large number of experimental hybrids and measure their performance across different environments to select hybrids with the highest yield. The best hybrids have been identified by trial and error. However, this process can take many years1. Research question of Syngenta Crop Challenge in Analytics 2019 was the following: Can environmental data be aggregated into useful metrics representing stresses encountered by maize throughout a growing season? Can these metrics be used to discriminate between hybrids tolerant and susceptible to the stresses they represent?Maize  and a relatively weak response to seasonal rainfall8. The dominant effect of EDD is increased vapour pressure deficit which drives faster transpiration rates in plants.Soil water deficits accompanied by excessively high temperatures are already indicated as the most probable yield limiting factor in maize production in the USA9, especially in a global warming world. Main indicator of drought is the lack of precipitation during some period of time and if that is happening in the growing season, it can result in abiotic stress since plants would not have enough water for the proper development. Frequently used extreme precipitation index indicates maximum number of consecutive dry days with precipitation  <  1 mm10. Another way to do that, i.e. indicate conditions with drought, is to calculate vapour pressure deficit (VPD)11. While in the mentioned paper the sum of VPD was calculated during the growing season, the average VPD can be calculated due to different growing season length in different environments.Drought is the most important environmental stress that limits crop yield12. Drought and heat stress at flowering can cause asynchrony in the tasselling and silking of maize13. It is clearly stated that the plant is sensitive to dry conditions during the pollination and the 4-weeks period after the tasseling is very critical14. Also, the stress during the grain filling period can affect yield at the end of the season15. Here we roughly divided growing season length (GSL) into four stages: 1) from planting date up to the tasseling (VT) approximately at 30% of GSL, 2) from tasseling to the end of blister stage (R2) approximately at 50% of GSL, 3) from milk stage (R3) to physiological maturity (R6) approximately at 75% of GSL, and 4) from physiological maturity to harvesting date. Percentages were estimated from the Pannar website16. Approximate duration of the maize growth stages was given for the average growing season length of 160 days from emergence to harvest, to which we added another 10 days from planting to emergence17.Maize growth stages and vulnerability of the plant to drought and heat stress during the specific stage can be found in the literatureIn order to answer the research question we engineered meteorological features to represent environmental stress throughout the growing season, developed the model for yield prediction, explained the contribution of the features to yield decrease and classified the hybrids based on their performance in the environments with different level of stress.For crop challenge 2019 Syngenta provided two datasets, performance and weather, related to their experimental sites with diverse environmental conditions across United States.Performance dataset encompassed 387,427 samples, which contained hybrid ID, yield (quintiles/hectare), environment ID as the identifier for the tested location  and year, planting date, harvesting date, irrigation and soil conditions, provided from 2008 to 2017. The soil variables included the percentage of sand, clay and silt, pH in soil, organic matter (OM), cation-exchange capacity (CEC), available water content (AWC) and saturated hydraulic conductivity of soil (KSAT). The number of hybrids in the experiment was 2,452 and the number of environments was 1,560.W/m2), maximum temperature (°C), minimum temperature (°C), precipitation (mm), vapour pressure (Pa) and snow water equivalent (kg/m2) for all environments. After removing samples with unknown irrigation ( ≈ 2%), removing some illogical samples where harvest date was before the planting date and averaging yield for the same hybrid and environment, remaining dataset consisted of 122,881 samples.Weather dataset consisted of the daily values of day length (seconds), solar radiation  listed several drought indicators and indices suitable for the analysis, but most of them need long term statistics, to be calculated. One of the indicators is hydro-thermal coefficient of Selyaninov (HTC), which is found to be more convenient for the growing season21. HTC presents the ratio between precipitation sum during the specific period of time and evapotranspiration, which is estimated as 1/10 of the sum of average temperatures > 10 °C, during the same period of time. Values between 0 and 1 correspond to dry conditions.From weather dataset we used the values of given weather parameters for each environment and sort them in the proper order. In climatology, indices of extreme temperatures commonly used in the analysis are tropical days  and tropical nights For each environment, we calculated meteorological features which indicate heat and drought stress, in four stages during the growing season Figure . Heat stThe predictive model of the yield was trained on the environmental level, based on the extracted meteorological and available soil features, with mean yield of each environment as the target variable. We calculated the mean yield for each of the 1,492 environments left after preprocessing step.24. It is an ensemble learning method based on a large number of decision trees, whose predictions are aggregated at the output level. Passing through trees is very fast since the computer only needs to go through a number of simple “if” statements. RF can deal with non-linear data and categorical features. It is robust to outliers and does not have issues with overfitting25. A total of 100 trees in the forest were considered. K-Folds cross-validation was used for performance evaluation. This means that the dataset was split into k = 10 folds, each of which served as the test set in one iteration and as a part of the training set in 9 others. Model results were evaluated with R2 score (coefficient of determination), root mean square error (RMSE) and Pearson correlation coefficient26.We used random forest (RF) as the regression model, which has been proven to work well in agricultural use-casesd is the dimension of each sample and N corresponds to the number of given samples. For each of N = 1492 environments we had eight features which described soil characteristics and 26 features for weather conditions, resulting in d = 34 features per each environment. We first reduced the dimensionality of our dataset using Principal Component Analysis (PCA) by keeping as many components as needed such that 99% of variance in the data is explained. In this way we projected our data, represented as matrix XN×d, onto a 9-dimensional subspace W ⊂ X, (WN×k with k = 9) in which we further applied GMM training procedure to estimate the density of our dataset. Therefore, the PDF for each environment w (row in the matrix WN×k) was modeled as M is the number of Gaussian components denoted with μ = {μj} stands for the centres of the components and corresponding covariance matrices Σ = {Σj}, while π = {πj} are the mixing coefficients (weights). The first problem that can arise in GMM training is related to the number of components M. The choice of M is a matter of trade-off between under- and over-fitting. Too small M will not account for the specifics in the dataset, while too large M will fit too closely to the data and loose the GMM’s ability to generalize. The second problem is that estimated covariance matrices might become singular during the training procedure. In order to overcome these issues we opted for a variational Bayesian GMM (VB-GMM) procedure27. VB-GMM allowed us to impose the prior to the model parameter π with a Dirichlet process. In practice, Dirichlet process inference algorithm is approximated and uses a truncated distribution with a fixed maximum number of components . Since VB-GMM determined the number of components M by setting some weights of the initial number of components very close to zero, we still needed this initial guess for the number of components. This was solved by fitting the expectation maximization GMM model without prior on our dataset by changing the number of components from two to 20 and calculating the Akaike information criterion (AIC) and Bayesian information criterion (BIC). Both information criteria estimate the relative amount of information lost by a given model, the less information a model loses, the higher the quality of that model. According to the obtained curves for calculated criteria we got two numbers of possible initial number of components in the model, one for which AIC criterion reached minimum value and other for which BIC criterion reached minimum value. We chose minimum of those two numbers (six in our experiments) in order to preserve model from overfitting  to enrich the data used for training the predictive model. GMM, defined as a convex combination of Gaussian densities, was used for modelling of the probability density function (PDF) of our dataset 31. The method that was used is general and could be applied to any classification or regression model to explain how features contribute to predictions. For every feature, we divided the range of the feature values from minimum to maximum into 10 bands, bounded by 11 equidistant values. Then, we examined the contribution of each of these feature’s values to the yield prediction. The GMM trained on the dataset was used to generate 10,000 surrogate samples, where each sample was given in the form of a feature vector. We then randomly chose 500 samples from the enriched dataset and altered the value of the feature that we wanted to test. We altered the value of this feature in all 500 samples to the desired value and calculated the difference between the predicted yield of original and altered samples. This analysis resulted in 500 yield differences, for which we calculated the mean and standard deviation. The procedure was repeated for all selected values and all features from the dataset.Model explanation was used to get a better insight in the contribution of the features in yield predictionThe next step in the research was to relate the performance of hybrids to different levels of heat and drought stress. But first, as the yield prediction model was applied on the environmental level, we needed to define stress metrics that will allow us to rank the environments according to heat and drought stress. To quantify how much stress was present in each environment we utilized previously calculated contributions of features. Feature value observed in the environment was approximated by the nearest from available 11 equidistant values ranging from feature minimum to maximum and corresponding contribution was derived. With this approach we were able to assign the contribution of each feature value to yield prediction. Next, we summed up the contributions of all features, separately for heat and drought, and proclaimed these values as the total amount of heat/drought stress in a particular environment. As a result, the environments were ranked based on the total amount of heat or drought stress present in them. Now the interaction of hybrids and environments could be translated to a new space defined by the environment stress level and hybrid yield. Hybrid performances were thus projected onto this space to further examine their tolerance to stress.32. Slopes  > 1 indicated a greater change in yield with changing environmental conditions, and slopes  < 1 indicated more stable yields across a range of environments, classifying the hybrid as stress tolerant. In multiple linear regression both slopes for heat stress and drought stress had to be  < 1 to classify the hybrid as tolerant to combined heat and drought stress. Samples where normal irrigation was applied were excluded from the stability analysis since it was not relevant to consider the yield at the certain environment if the water deficiency was overcome and the stress was reduced33.To characterize the response of a hybrid to stress across all the environments where it was grown, linear regression was performed between the yield and the level of stress at the environments, for heat and drought separately. On the other hand, multiple linear regression was used to examine the combined effect of both heat and drought stress. The slope of the regression was used as a measure of yield adaptability2 score = 0.33, RMSE = 18.0 quintiles/hectare and Pearson correlation coefficient = 0.58 showed moderate correlation between them  not to reduce dataset for 22.11%, b) to examine contribution of heat and drought stress features to the yield across all environments and c) to examine performance of the hybrids resistant to all stresses, in optimal conditions, where normal irrigation was applied.The results of RF regressor were evaluated by comparing observed and predicted yield. Obtained results with Rhem Fig. . After lThe main output of model explanation was the contribution of each feature to the yield prediction. The contribution was expressed as the mean value over 500 iterations. Example of the contribution of two features representing stress metrics EDD3 (heat) and PS2 (drought) is presented in Fig. The mean contributions of all features, obtained through the model explanation, had both positive and negative values. We provide an example in Table 34 and that lower yields can be realized in dry conditions on the soil with higher clay content compared to the management zones with lower clay content35 while this phenomenon appears reversed in wet conditions. Texture classes between 10–30% of clay content had air and moisture regimes that are optimal for healthy maize production  atmospheric reanalysis of the global climate. Reanalysis combines model data with observations from across the world into a globally complete and consistent dataset using the laws of physics, in the process called data assimilation. Horizontal resolution of hourly data is 0.1° × 0.1°, with the temporal coverage from 1981 to present.This methodology is scalable and it can be applied for any region of the world with proper agricultural data, since soil data are available from SoilGridsSupplementary Information.


Dendrite growth poses a significant problem in the design of modern batteries as it can lead to capacity loss and short‐circuiting. Recently, it has been proposed that self‐diffusion barriers might be used as a descriptor for the occurrence of dendrite growth in batteries. As surface strain effects can modify dendritic growth, we present first‐principles DFT calculations of the dependence of metal self‐diffusion barriers on applied surface strain for a number of metals that are used as charge carriers in batteries. Overall, we find a rather small strain dependence of the barriers. We mainly attribute this to cancellation effects in the strain dependence of the initial and the transition states in diffusion. To strain or not to strain: We calculate the strain dependence of metal anode surface properties from first principles to explore whether surface stress can be related to dendrite growth in metal‐ion batteries on the atomistic level. We only find a weak dependence of these properties on the applied strain, which suggests that strain does not influence dendrite growth on the atomistic scale, but rather dendrite‐related morphologies on longer‐length scales. Since their introduction in 1991, Li‐ion batteries have become the dominant energy storage technology for portable devices and have seen various improvements over the years, which has allowed them to approach their theoretical energy density.+, HF, and LiNO3, however, unfortunately these electrolyte additives are usually consumed in side reactions.Still, much like their predecessor, post‐Li batteries suffer from a variety of technical problems, arguably the worst of which is the lack of long‐term stability of the electrodes.It is important that any modifications of the SEI do not compromise the desired properties of the SEI. In particular, a high ion conductivity and, consequently, a high number of diffusion pathways are necessary to ensure a uniform surface growth distribution.Previously, we suggested that the height of self‐diffusion barriers could be used as a descriptor for dendrite growth in batteries.>20 Å. A Γ‐centered 5×5×1 k‐point grid was used to calculate the energies. The electronic self‐consistent‐field (SCF) scheme was converged up to 10−5 eV by using the Methfessel–Paxton smearing scheme−4 eV. In our calculations, we considered a maximum strain of ±3 %, as larger strains in metal films are typically released by the formation of dislocation networks.We performed the calculations in this work by applying DFT using the plane‐wave‐based Vienna ab initio simulation package (VASP).Ecoh were determined by subtracting the energy of the isolated atom in a vacuum Evac from the corresponding bulk energy Ebulk per atom [Eq. (1)]:We will first present the results for the different metal systems without any applied strain and then discuss the strain effects for various properties of the metal surfaces. The metal cohesive energies k‐point density did not improve the cohesive energy of Zn. The underestimation of the Zn cohesive energy in PBE‐DFT calculations has been found before.The lattice constants were derived from the minimum of the cohesive energy. Our calculated results are compared with experimental data in Table Esurf is defined as the energy per area required to form a specific surface from a bulk structure. Here Esurf was calculated by employing two approaches. For a symmetric slab in which the topmost layers on both sides of the slab are relaxed, the surface energy is given by Equation The surface energy is a measure of the energy cost to create a particular surface and can be used to estimate the likely surface terminations of a given metal. The surface energy Esymrelaxedrepresents the energy of the relaxed surface system, Natom is the number of atoms per super cell, and A is the surface area of the supercell. These symmetric slab calculations usually require relatively thick slabs that lead to a larger computational effort. However, surface energies can also be derived for thinner asymmetric slabs in which only one side of the slab is relaxed and the other side is kept at its ideal bulk positions according to Equation in which Easymrelaxedrepresents the energy of the slab with one relaxed surface and Estatic is the energy of the static slab in the ideal bulk termination without any relaxation. We compared the surface energies obtained from both approaches to check the reliability of the results, as any significant disparity between the results would indicate an insufficiently large slab size. For all considered metals we calculated the surface energies of the most stable surface terminations with a significant area fraction within the Wulff construction.in which Both the symmetric and the asymmetric surface slabs were calculated by using seven‐layer slabs with either the inner three layers or the bottom five layers kept at the ideal bulk positions. Both approaches yield very similar results, which indicates that the calculated surface energies are converged with respect to the slab thickness. We list the results for the asymmetric calculations in Table To study strain effects, we changed the lattice constants within the range of −3 to +3 %. We did not consider possible phase transitions between different lattice structures induced by strain, as they can occur for Li and Na under high‐pressure conditions.<0.05 J m−2) with the exception of the Li(1 1 0) surface (>0.08 J m−2). Interestingly, the most densely packed (1 1 1) surfaces and the open (1 1 0) surfaces exhibit the same trend in the surface energies as a function of lattice strain, whereas the (1 0 0) surfaces, which are intermediate with respect to surface roughness, show the opposite trend. Hence there is no simple explanation for the observed trends.If we compare the resulting data for the alkali metals shown in Figure −2.The direct comparison between the different alkali metals Figure  shows a We calculated the metal adsorption energies for the most stable adsorption sites for all considered surfaces according to Equation E/slabatom is the energy of the slab with one adsorbed atom per surface unit cell, Eslab is the energy of the clean slab, and Evac corresponds to the total energy of the isolated metal atom. All adsorption calculations were performed under the same conditions using two relaxed surface layers.in which To illustrate the observed trends in the metal adsorption energies, the adsorption energies on Li(1 0 0), Li(1 1 0), Na(1 1 0), and Mg(0 0 0 1) are plotted as a function of lattice strain in Figure Notably, on late transition metals, typically, stronger binding on expanded surfaces has been observed.We used the calculated adsorption energies to estimate the terrace diffusion barriers by Equation Etrans represents the energy of the transition state and Emin refers to the adsorption energy on the most favorable adsorption site. We emphasize that we only consider hopping diffusion processes on terraces in this work. So‐called exchange processes can be more favorable at step sitesin which We summarize the change of the diffusion barriers as a function of the lattice strain in Figure The weak dependence of diffusion barriers on lattice strain might come as a surprise if we consider that some adsorption energies show a strong dependence. However, one has to take into account that diffusion barriers correspond to the difference of the energies of the transition state and the initial adsorption state. Both of these energies are influenced by surface strain, but a similar dependence of both energies on the strain leads to a cancellation effect and results in a weak variation of the height of diffusion barriers with lattice strain. A similar phenomenon has been found for reaction barriers in methanol oxidation in heterogeneous catalysis on Cu(1 1 0).Although the lattice strain does not seem to affect the diffusion barriers along the minimum energy path between the most favorable adsorption positions strongly, it does affect a property of the potential energy surface (PES) that we call roughness. The roughness of the PES can be defined as the energy difference between the least and the most favorable adsorption site on the surface [Eq. (6)]:Notably, we did not explicitly sample the whole surface to find the energetically least favorable position, we selected the positions with the highest and lowest adsorption energy from our sampling of the high‐symmetry surface sites. The surface roughness can be of interest as it quantifies the maximum variation in the PES that is still relevant for the mobility on the surface. A very smooth PES would indicate high mobility, whereas a surface with a high PES roughness can be associated with slower diffusion. Such a distinction is not possible by only taking the primary diffusion process into account. The effect of strain on the PES roughness was more pronounced than the effect on the minimum diffusion barrier and also more monotonic. We observe a general trend in the PES roughness Figure . For neaWe studied the influence of strain on several surface properties of metals that are used as charge carriers in batteries. The variation in the surface energies correlates with the cohesive energy of the respective metals. For almost all surfaces, the surface energy was decreased by strain effects. However, there is no universal preference for compression or expansion with regard to the decrease of the surface energy as it is dependent on both the surface termination and the element in question.The strain dependence of the adsorption energy is even less universally predictable as it is dependent on the adsorption position, the surface termination, and the particular metal. No correlation of the strain dependence of the adsorption energies with the cohesive energy of the metal was found. Some sites even exhibit a completely disproportionate dependence on the strain compared to other sites of the same element and surface.As a result of the lack of clear trends and rather strong variations in adsorption energy for specific sites, the strain effects in adsorption can apparently only be captured on a case‐by‐case basis. Strain‐related effects on the self‐diffusion barriers are rather small for nearly all of the metals and surfaces considered. This can be attributed to cancellation effects in the strain dependence of the initial and the transition states in diffusion. Only the overall roughness of the potential energy surfaces for adsorption exhibits a linear dependence on the strain for a number of metals. Therefore, it seems reasonable to conclude that the strain‐induced changes observed experimentally in the growth behavior of dendrites are not caused by changes in the atomic transport but rather because of the effect of strain on the morphology of the growing dendrites on a larger‐length scale and on the structural properties of electrode–electrolyte interfaces and solid–electrolyte interphases. To substantiate this, however, requires more detailed studies that also consider the electrochemical environment.The authors declare no conflict of interest.As a service to our authors and readers, this journal provides supporting information supplied by the authors. Such materials are peer reviewed and may be re‐organized for online delivery, but are not copy‐edited or typeset. Technical support issues arising from supporting information (other than missing files) should be addressed to the authors.SupplementaryClick here for additional data file.


Bayesian networks are powerful statistical models to understand causal relationships in real-world probabilistic problems such as diagnosis, forecasting, computer vision, etc. For systems that involve complex causal dependencies among many variables, the complexity of the associated Bayesian networks become computationally intractable. As a result, direct hardware implementation of these networks is one promising approach to reducing power consumption and execution time. However, the few hardware implementations of Bayesian networks presented in literature rely on deterministic CMOS devices that are not efficient in representing the stochastic variables in a Bayesian network that encode the probability of occurrence of the associated event. This work presents an experimental demonstration of a Bayesian network building block implemented with inherently stochastic spintronic devices based on the natural physics of nanomagnets. These devices are based on nanomagnets with perpendicular magnetic anisotropy, initialized to their hard axes by the spin orbit torque from a heavy metal under-layer utilizing the giant spin Hall effect, enabling stochastic behavior. We construct an electrically interconnected network of two stochastic devices and manipulate the correlations between their states by changing connection weights and biases. By mapping given conditional probability tables to the circuit hardware, we demonstrate that any two node Bayesian networks can be implemented by our stochastic network. We then present the stochastic simulation of an example case of a four node Bayesian network using our proposed device, with parameters taken from the experiment. We view this work as a first step towards the large scale hardware implementation of Bayesian networks. This work focusses on a compact hardware that can generate random numbers with controllable probability by using its intrinsic physics and can serve as efficient building block for Bayesian networks.There has been increasing demands to realize specialized hardware for unconventional computing tasks where software based approaches running on general purpose CPU cannot efficiently execute the basic steps of the associated stochastic algorithms, such as generating random numbers. For example, a single random number generator based on CMOS linear feedback shift register requires around 1000 transistors2. In a BN, each node represents a stochastic variable, whose probability of occurrence is determined by the states of its parent nodes. The dependence between a set of such nodes is given by a conditional probability table (CPT). BNs are traditionally implemented in software aiming at applications in areas such as forecasting, diagnosis, and computer vision3. However, as the complexity of the BNs grows, i.e., as the number of parent nodes affecting the probability of a particular child node becomes large, both the assessment of that child node probability, and the inference about the possible cause becomes impractical4. Specifically, as the network size grows, the number of terms in the calculation of the joint probability using probability chain rule increases rapidly4.Bayesian networks (BNs) are directed graphical models that are used to represent the causal dependencies among stochastic variablesDirect representation of Bayesian networks in hardware has been proposed as an alternative way to perform the two above mentioned tasks, i.e., probability assessment and inference. In this case, each “node” in a Bayesian network is represented by a stochastic device, having a distinct probability of being in one of two possible states. This probability is controlled by the input it receives dependent on the states of its parent nodes, through the weights of the connections between them. The CPT is encoded in the weights of these connections. By representing a BN with a hardware network of this kind, the required probability of a particular event is readily obtained by sampling the output of the corresponding stochastic device. Moreover, inference about the possible cause of a particular event can be evaluated by observing the joint distribution of the two stochastic devices corresponding to the “event” node and the particular “cause” node of interest.5 demonstrated FPGA based BN implementation utilizing suitable architectural design and memory allocation schemes. Cai et al.6 demonstrated another FPGA based architectural design along with a suitable pseudo random number generator. Manisinghka et al.7 implemented a BN in digital circuits using a novel abstraction. Chakrapani et al.8 and Weijia et al.9 proposed a probabilistic CMOS hardware for BN implementation, however there has not been an experimental demonstration in literature to our knowledge. Nevertheless, there is an interest in a compact implementation of the stochastic nodes of a BN and their conditional relations.Several hardware implementations of BNs have been proposed based on CMOS hardware. For example, Zermani et al.In this work, we present an experimental demonstration of a spintronics based compact hardware implementation of BNs. The stochastic elements are implemented naturally by a compact device consisting of a perpendicular nanomagnet. The CPTs are translated directly to the connection weights, implemented by resistive connections between such devices.22 and are given the name “p-bit”, which is the short form of “probabilistic bit”. It has been proposed that inherently unstable nanomagnet can be a natural implementation of the stochastic variable in a BN23. We first present a p-bit implementation using a stochastic spintronic device that has isolated input and output to allow for interconnection in circuitry. The output of such a device is a tunable random number, whose mean is controlled by an electrical input. Then, we build an electrically connected network of two such devices and study the correlation of their outputs for different connections and biases. We show that any CPT can be implemented by changing the connections and biases of this circuit, thus representing a hardware BN building block. Finally, using parameters taken from the experiment, we perform a stochastic Landau Lifshitz Gilbert (sLLG) simulation of a four node BN and compare the results of the forecast with those expected from calculating joint probability distributions.Unstable nanomagnet based spintronic devices have recently attracted much research interest for probabilistic spin logic (PSL)18. The top left of Fig. 25. However, when the spin current density is large enough, and when this field is absent, the magnetization gets pinned in the direction of the spin polarization, i.e. the magnets hard axis. Once the spin current is removed, ambient thermal noise makes the magnetization relax to either “up” or “down” with equal probability due to the symmetric energy landscape for these two states26 as depicted by the cartoon in the top right of Fig. OUT is + ve for a magnetization in the “up” direction and −ve for “down”. The probability of relaxing back to the “up” or “down” direction can be controlled by applying a small out-of-plane magnetic field that lifts the degeneracy of the energy landscape. A positive field in the z-direction lowers the energy of the “up” state and raises that of the “down” state, thus making the “up” state more favorable. A negative z-directed field does the exact opposite. This is depicted in the energy landscape diagrams shown in the bottom panel of Fig. In our experiment, the stochastic device is based on a hard axis initialized magnet with perpendicular magnetic anisotropy (PMA), whose output probability is controlled by the magnetic field produced by a charge current passing through an isolated metal ringS, HK and damping. The bottom panel of Fig. Figure mentclass2pt{minimFor experimental implementation, starting with a stack of Ta(5 nm)/CoFeB1 nm)/MgO(2 nm)/Ta(1 nm) thin film, a Hall bar device with a PMA magnetic island located at the center is fabricated by means of successive e-beam lithography and Ar ion milling steps. To generate the out-of-plane field for tunability, the “Oersted ring” is fabricated on top and electrically isolated from the Hall bar by a dielectric layer. A false colored SEM image of the fabricated device is shown in the inset of Fig.  nm/MgO taken to match the sigmoidal activation function obtained from the experiment.We first demonstrate that the CPT between the two probabilistic random variables in our example can be implemented by design of proper electrical connections between two of our stochastic devices ,,4), we oThe parameters Figure In our circuit as shown in Fig. Note that 2 in Eq. .Next, we show that the same circuit can capture any given CPT, by changing the entclass1pt{minima1 in Eq.  sequentially under the influence of all the other nodes, which are fixed to their current values.We take five different CPTs with “ to Eqs.  and 8)..\documenentclass1pt{minimaentclass1pt{minimaN where N is the number of parent nodes for the particular child node of interest4. Instead of performing this algebra, the required probability can be obtained from the circuit by directly observing the stochastic output of device 2 and obtaining its mean value over several pulsing cycles. This luxury of having to observe only the nodes of interest while disregarding all the other nodes is an advantage of using a probabilistic algorithm, versus calculating the probabilities using deterministic rules as discussed by Feynman31 and utilized in many sampling schemes27.The probability of finding a stale cheese block can be found from the joint probability distribution by using the probability chain rule:Similarly, given that a randomly drawn cheese block from a large lot is stale, the probability that it was caused by a low quality packaging material can be found by using Bayes theorem:N where N is the number of potential binary causes of a particular effect4. However, from the hardware BN, this probability can be directly obtained by observing the joint distribution of states of the two devices. It is to be noted here that this way of performing the inference always involves observing the joint distributions of only two nodes of the BN: nodes corresponding to the effect and the potential cause of interest, irrespective of N.The number of terms required in the evaluation of the above expression also grows as ~ 2In our experiment, after 100 pulsing cycles, the obtained output probabilities for all the five circuits ..\documenThe coupling and bias component of  in Eqs.  and 8)  \documenWhile solving the coupled sLLG, each p-bit is put along the hard axis by the GSHE current in a sequential order from parent to child node and the magnetizations of all p-bits are recorded after their corresponding pulse is turned off. It is worth noting that the pulse sequence is important for the proper operation of the Bayesian network. The pulsing should start from the first node and move down the hierarchy from parent to corresponding child nodes. The order of pulsing among different nodes on the same hierarchy level (e.g. node mentclasspt{minimafrom Eq. . It is t7 or RAM-based Linear Feedback Gaussian Random Number Generators6 that require a large area footprint. What we have demonstrated here is a compact true random number generator (TRNG) capable of operating at few hundreds of MHz. The speed of producing statistically correct random numbers by the device and hence the latency of the network is limited by the time required for SOT hard axis initialization and magnetization relaxation after removal of SOT, which is shown to be around 30 ns in the sLLG time plot panels of Fig. 15. Compared to previously demonstrated spin based TRNG35, this implementation employs a different scheme to generate random numbers. In our approach, any applied current that is larger than that required for hard axis initialization of the magnet will result in the generation of a random number with the correct statistic once the current pulse is removed23. Hence, in a large network, the device to device variation in the required current can be easily mitigated by choosing the largest value of the required current among all devices. Possible variations in the shape and offset of the sigmoidal activation function of our devices can be controlled by appropriately choosing the parameters B0 and h2 while translating the given CPTs into the connection weights, shown in Eqs.  effect for hard-axis initialization as proposed by Scott et al.38 in their benchmarking study (section IV of the main text). In addition, employing magnetic tunnel junctions (MTJs) instead of AHE can eliminate the need for OP-AMPs for readout. The typical difference in the two stable resistive states of an MTJ is of the order of 10 kΩ, whereas in case of AHE, it is a few ohms for standard material stacks. This allows the elimination of the OP-AMPs for readout. Implementations of an MTJ based readout scheme, where the state of the free layer magnet is converted to a voltage by a potential divider formed by the MTJ and a normal resistor was presented by Camsari et al.13 (figure 3 of the main text) and Hassan et al.39 (figure 4 of the main text). In these references, the voltage swing generated at the output is large enough to be converted to a “rail-to-rail” swing by a single inverter. In the above references, the MTJ free layer was designed to be a low barrier magnet, but the analysis of the output swing remains unchanged for our hard axis initialization scheme with stable magnets. The power dissipated for reading the MTJ resistive state and amplifying the output signal through the inverter would be the same for the device presented in this study when scaled to similar dimensions. The additional power required for hard axis initialization of the nanomagnet is similar to that required for deterministic switching in SOT-MRAM applications as the required currents are comparable40. This power requirement could be much smaller if the hard axis initialization is done via VCM, as evaluated by Scott et al.38 (Table 1 of their main text). Finally, the tunability obtained through Oersted field in this work can be replaced by a more efficient STT mechanism, which consumes similar currents as required for hard-axis initialization41.Previously proposed hardware implementations of Bayesian networks have used CMOS based pseudo random number generators realized with XOR-SHIFT circuits in Eqs. . Also noWe have experimentally demonstrated that by connecting two stochastic spintronic devices and designing the connection and bias parameters, BN building blocks can be implemented in hardware. By implementing BNs using such hardware, both probability assessment and inference can be performed by sampling the output of only the relevant nodes. Using experimentally benchmarked sLLG simulations, we have shown that a four node BN implemented in hardware using the presented stochastic devices can generate probabilities that are well matched to the theoretical values from calculating the joint probability distribution. This demonstration serves as a step towards building large scale hardware systems for implementing Bayesian networks.


NEE is the difference between the total CO2 release due to all respiration processes (RECO), and the gross carbon uptake by photosynthesis (GPP). These two gross CO2 fluxes are derived from EC measurements by applying partitioning methods that rely on physiologically based functional relationships with a limited number of environmental drivers. However, the partitioning methods applied in the global FLUXNET network of EC observations do not account for the multiple co‐acting factors that modulate GPP and RECO flux dynamics. To overcome this limitation, we developed a hybrid data‐driven approach based on combined neural networks (NNC‐part). NNC‐part incorporates process knowledge by introducing a photosynthetic response based on the light‐use efficiency (LUE) concept, and uses a comprehensive dataset of soil and micrometeorological variables as fluxes drivers. We applied the method to 36 sites from the FLUXNET2015 dataset and found a high consistency in the results with those derived from other standard partitioning methods for both GPP (R2 > .94) and RECO (R2 > .8). High consistency was also found for (a) the diurnal and seasonal patterns of fluxes and (b) the ecosystem functional responses. NNC‐part performed more realistic than the traditional methods for predicting additional patterns of gross CO2 fluxes, such as: (a) the GPP response to VPD, (b) direct effects of air temperature on GPP dynamics, (c) hysteresis in the diel cycle of gross CO2 fluxes, (d) the sensitivity of LUE to the diffuse to direct radiation ratio, and (e) the post rain respiration pulse after a long dry period. In conclusion, NNC‐part is a valid data‐driven approach to provide GPP and RECO estimates and complementary to the existing partitioning methods.The eddy covariance (EC) technique is used to measure the net ecosystem exchange (NEE) of CO C‐part) for partitioning net ecosystem exchange of CO2, measured by eddy covariance over land ecosystems, into ecosystem respiration (RECO) and gross primary production (GPP). NNC‐part has a hybrid nature for assimilating the light‐use efficiency (LUE) concept. The GPP and RECO produced by NNC‐part resulted consistent with estimates provided in FLUXNET2015 and patterns of fluxes very realistic, being the method able to reproduce: the GPP and RECO response to meteorological forces; hysteresis in the diel cycle of fluxes; effect of diffuse to direct radiation ratio on LUE, respiration pulse.In this paper, we present an innovative neural network (NN GPP is the gross amount of carbon uptake by photosynthesis from vegetation while RECO is the total carbon efflux by the respiration processes of all organisms. Estimating GPP and RECO is a key step to better understand the underlying mechanisms constraining ecosystem function. Moreover, GPP and RECO estimates from EC are useful for modeling, supporting process‐based model parameterization and validation, data assimilation, plant trait retrieval by model inversion  have been proposed for estimating GPP and RECO from NEE measurements , their direct application at the ecosystem spatial scale should be evaluated carefully Medlyn, . Further2 and H2O fluxes dynamics in the flux variance similarity approach for simultaneously partitioning carbon and water fluxes. The method, although interesting, requires canopy scale estimates of water‐use efficiency (WUE), which introduces assumptions and uncertainty. More recently, the estimation of gross CO2 fluxes from NEE has been approached using the EC method in combination with parallel measurements of a trace gas such as carbonyl sulfide  and output (fluxes) are inferred from the data. Several studies have reported the capability of machine learning to reproduce complex patterns in ecological studies, and in particular in relation to EC measurements , which have already been tested as a partitioning tool in a cross‐site intercomparison among several partitioning methods  the functional relationships between micrometeorological inputs and fluxes in output as reproduced by the ANN; and (d) the effectiveness of the proposed method to predict additional patterns of gross CO2 fluxes not (or less) accounted for by the standard partitioning methods.Our aim was to design a machine learning‐based method flexible enough to: (a) use a large set of drivers, accounting for the complexity of the GPP and RECO responses, (b) ensure general applicability of the method to allow ANN‐based partitioning in the context of FLUXNET thanks to the use of generally measured variables. In order to evaluate the results, we compared the GPP and RECO produced by the ANN with the estimates obtained by two partitioning methods used in FLUXNET, by analyzing: (a) the consistency of the estimates at yearly, seasonal, daily, and  hourly time steps; (b) the dynamics of the seasonal and diurnal cycles of the estimated gross CO22.1m neurons in one layer are the inputs for n neurons of the next layer and the signals are transferred through connections associated with multiplicative weights. The learning (or training) process of an ANN consists of adjusting the weights of the network on the basis of specific examples provided as input (supervised training).ANNs are nonlinear and nonparametric methods for regression and classification that artificially emulate the functioning of a biological brain Haykin, . The bas1The use of this term is not incidental: surrogate modeling, also known as emulation, is about developing statistical models that learn to mimic costly physical models, such as radiative transfer or climate models, using a representative dataset of simulation/forward runs. ML algorithms learn such parameter–observation relation with high accuracy, and after training can be used for forward simulation very efficiently  is based on the concept that NEE, measured by the EC system, is the difference between RECO and GPP (Equation RECO) and the other for retrieving GPP (SNNGPP). The output of these two subnetworks  is combined in the last neuron of the overall structure, where NEE is calculated and compared with the measurements in order to optimize the network's weights (network training). The output signals of both subnetworks are constrained to be always positive, except for SNNGPP, which is constrained to be 0 during nighttime, as photosynthesis requires light. However, RECO and GPP have opposite signs; therefore, the connection weight of RECO is fixed to be positive (wRECO = 1) and that of GPP to be negative (wGPP = −1), mirroring the sign convention of the measured NEE. Since the last transfer function is unbounded and linear, and the bias is fixed to 0, the last node effectively reproduces the equation of NEE as:In this experiment, we developed a new ANN architecture designed to emulaten hidden neurons (different for the two fluxes) and a hyperbolic tangent sigmoid activation function (“Tansig”), which improved the performance of network optimization; the second has only one neuron and a log‐sigmoidal activation function (“Logsig”) that constrains the output to positive values, allowing the implementation of Equation ; thus, the output from the SNNGPP is 0 at nighttime and positive during the day. With this structure, the last node of the SNNGPP mimics the light‐use efficiency (LUE) approach: the output coming from the two previous hidden layers is an LUE that is then multiplied by the available shortwave radiation to estimate GPP. In this way, the drivers are used to estimate an instantaneous proxy of the LUE which takes into consideration seasonal and diurnal variability of carbon uptake, including photosynthesis saturation at high light, while shortwave radiation mainly defines the magnitude of the flux. This decreases the weight given to the incoming shortwave radiation in relation to other driving variables. Up to this step, the output of the two subnetworks (SRECO and SGPP) have positive signs; the sign convention of the NEE equation is finally restored in the last neuron of the overall structure by associating a positively signed weight to SRECO and a negatively signed weight to SGPP.The two subnetworks have different input drivers but a similar structure including two hidden layers: the first hidden layer has Equation  in the l2.2http://fluxnet.fluxdata.org) and, as an additional comparison, a synthetic dataset generated by the MUlti‐layer Simulator of the Interactions between a vegetation Canopy and the Atmosphere (MuSICA) process‐based model  NEE observations (assumed as representative of RECO given the absence of photosynthesis at night) to fit the Lloyd and Taylor equation. There are two parameters to fit in the Lloyd and Taylor equation: the activation energy (E0) and the respiration rate at the reference temperature (Rref). In the NT method, E0 is estimated at an annual scale by calculating E0 values every 15 days and then averaging the three with smaller uncertainty. Once E0 is fixed, the Rref parameter is estimated using short‐term moving windows (4 days). On the basis of Equation  from Reichstein et al., ref, are fitted using daytime measurements. In DT, the light response curve is driven by the incoming shortwave radiation (SW_IN), but it is adjusted for the effect of stomata closure due to the atmospheric evaporative demand using VPD as an additional driver of GPP  measured with the EC technique. In particular, we used the NEE_CUT_USTAR50 variable , VPD (kPa), air temperature (TA [°C]), soil temperature (TS [°C]), and soil water content (SWC [%]) in the upper soil layer  were also used in order to provide information about seasonality and length of the day. The SW_IN_POT was also aggregated at daily resolution to better track the seasonality of light conditions; the first derivatives of half hourly and daily SW_IN_POT were also calculated to add specific information about the seasonal and diurnal dynamics of light  and a proxy of GPP (GPPprox) were calculated and used in input. In this case, gap‐filled  as follow:k is the fraction of daytime hours for each day.Finally, since plant photosynthesis and RECO change seasonally also due to substrate availability and management (in case of managed sites), daily average of nighttime NEE . To take this into consideration, we used their sine and cosine transformations instead of the original variable's value.RECO used TA, TS, SWC, WS, NEENIGHT, and sine and cosine of WD and DOY while the drivers for the SNNGPP were SW_IN, SW_IN_POT , VPD, TA, SWC, WS, GPPprox, and sine and cosine values of WD. Note that SW_IN is then used both as driver and as single input in the GPP subnetwork for the product with the LUE XnormAvailable records with all the inputs and the measured output were randomly split 25 times in sets for training , test (20%) to avoid overfitting, and validation (20%) to assess the performances. This results in 25 datasets used for the ANN training. The ANN training used the mean square error as the cost function and the Levenberg–Marquardt backpropagation algorithm.RECO and SNNGPP, respectively. For each one of the 25 training datasets, the five networks with different structures were trained five times, changing the initial weights randomly, and then selecting the one with the best performance on the validation set. In case the sign of RECO and GPP did not respect the expected convention, the weights were reinitialized and the training was repeated. At the end of the process, there were 125 trained ANNs for each site‐year (25 training datasets × 5 structures). To select the best among them, which were then used for the application, the ANN with the highest performances on the validation set was selected for each training dataset (obtaining 25 selected ANNs), and then, the five ANNs with the best model efficiency in predicting the overall NEE time series were finally selected and applied. RECO and GPP were calculated by averaging the output of the two subnetworks SNNGPP and SNNRECO from the five selected ANNs. The capacity of the ANN to simulate NEE was also analyzed . The GPP and RECO estimates from the different methods were compared at half hourly, daily, and yearly time resolutions and were further evaluated about the consistency among methods for the seasonal cycle and its daily anomalies. The seasonal cycle was estimated by averaging the fluxes over 5 days while the daily anomalies were estimated by subtracting the seasonal cycle from each daily value. Finally, in order to compare the seasonal cycles across sites‐years having different flux magnitudes, the annual average daily value was subtracted from each seasonal cycle. In addition to the seasonal cycles, the mean diurnal cycles were calculated and compared. To verify that the different drivers used in the NNC‐part had the role that is ecologically meaningful, functional relationships between micrometeorological drivers and retrieved gross CO2 fluxes were analyzed. In particular, the functional relationship between SW_IN and GPP, VPD and GPP, and TA and both GPP and RECO were considered and analyzed. This was carried out in two different ways: (a) we simulated the flux responses in the NNC‐part varying artificially one driver at time, while the other drivers were kept fixed to the average condition of the season at midday; and (b) we compared the seasonal variations of gross CO2 fluxes with respect to the meteorological forces for both fluxes calculated using the NNC‐part results and those obtained using the NT and DT methods. The responses were obtained by averaging the values of 10 equal intervals step of the fluxes and drivers for 2 months.Results of this experiment were evaluated by comparing GPP and RECO from NN2 fluxes that are not explicitly accounted for by the DT and NT methods but can be indirectly considered in the NNC‐part method: (a) the instantaneous LUE responses to the diffuse to direct radiation ratio, (b) the hysteresis of the diel cycles of predicted RECO and GPP with respect to TA and SW_IN, and (c) the respiration pulse after a rapid increase in SWC occurring frequently in dry environments. Finally, the NNC‐part algorithm implemented here allows for an analysis of the role and importance of the different drivers  with the DT method for GPP estimation and with the NT method for RECO, in both cases higher than the correlation between DT and NT methods . Looking at the bias, the magnitude of both GPP and RECO retrieved by NNC‐part was slightly lower than the DT and NT methods, with the predicted values by NNC‐part closer to the DT method . The average values of bias among methods were low on average (<27 g C m−2 year−1) and comparatively lower with respect to the NEE random uncertainty reported in the FLUXNET2015 database (<10% of the reported “NEE_CUT_USTAR50_RANDUNC”). The agreement between NNC‐part and the two FLUXNET standard methods was significantly lower in the case of RECO suggesting that the differences occurred in the diurnal and seasonal cycles of RECO between NNC‐part and both DT and NT methods were comparatively larger than GPP.The half hourly GPP and RECO retrieved by NNds Table . The meaC‐part and the DT and NT methods in both GPP and RECO were found for daily, seasonal, and yearly values and also for the daily average anomalies with respect to the seasonal values , in particular for GPP (R2 > .97). In terms of bias, results by NNC‐part were more consistent with the DT results confirming the findings from the subdaily comparison. High consistency was also found in the case of GPP daily anomalies (R2 > .88 and RMSE < 0.52 µmol CO2 m−2 s−1), but it decreased in the case of RECO  was confirmed by comparing the gross CO2 fluxes derived by training the methods on synthetic NEE  all methods slightly underestimated both GPPMUSICA and RECOMUSICA (bias ranging between −0.2 and −0.45 µmol CO2 m−2 s−1) with the largest discrepancy found in the case of DT method  during the growing season  and in the late afternoon; instead, the GPP estimated by NNC‐part was slightly higher than DT from the morning until midday, where NNC‐part estimated higher GPP values. A similar pattern appeared when comparing the two standard methods  although shifted toward positive values, indicating higher values of GPP predicted by the NT method compared to the NNC‐part. The GPP derived from NEEMUSICA confirmed the differences we highlighted between DT and the other methods  during the morning; the differences between DT and NT fitted on NEEMUSICA were also enhanced. These differences increased during the nighttime hours, while the predicted RECO by the two standard methods became closer during the daytime hours . It is interesting to note that the peak of the maximum RECO by NNC‐part was shifted compared to the DT and NT methods and closer to the one of RECOMUSICA, suggesting a better capacity to reproduce diurnal patterns when multiple drivers were involved  as opposed to the other methods, although the pattern of response was basically the same. This could be an indication of a water resources limitation during the summer period and at higher temperatures, that could be detected by the NNC‐part since it also used SWC as a driver. In fact, the functional relationships between RECO and SWC retrieved by NNC‐part highlighted a clear direct relationship between SWC and RECO in drought conditions, in particular for non‐forested ecosystems  that was reported in Figure The diurnal cycle of gross CON Figure a–c and tN Figure e–g. We gC‐part detected an appreciable hysteresis in the diurnal cycle  to changes in SWC  is in general difficult due to the sporadic nature of the events and the noise associated with the measurements. We evaluated the performances of the three models in a specific event that was clearly visible in the NEE time series . The pattern of the GPP–VPD relationship captured by NNC‐part is consistent with current knowledge about the protective  mechanisms of stomata closure carried out by plants in response to the increase in atmospheric evaporative demands  is in late spring/early summer, with lower fluxes of daytime RECO predicted by NNC‐part compared to both standard methods. When evaluated on the synthetic dataset, the mismatch between NNC‐part and the standard methods was significantly lower except for the anomalies, which were better reproduced by NNC‐part. In addition, the estimates from NNC‐part better matched the diurnal dynamics of RECOMUSICA than the estimates from the standard methods  without any relevant effect on NEE. This could be related to a certain limitation of these methods to infer the complex relationships of the MUSICA model with the reduced set of drivers used in this experiment, but we have also to consider that both NNC‐part and standard methods of partitioning were trained on noisy NEE signals  and the relationship between inputs and outputs is set on the basis of the data without any assumption . This property could be a great advantage when used to study a phenomenon that does not respond with the same invariant pattern. The synthetic datasets used in this experiment are generated using functional relationships which are more complex than the ones implemented in the FLUXNET standard methods. The better matching by NNC‐part highlights the potential of this algorithm to capture additional patterns of carbon fluxes dynamics. As a direct consequence, this led to some systematic differences observed in the GPP and RECO patterns compared to standard methods. For instance, the shape of the GPP diurnal cycle predicted by the DT method is almost symmetrically bell shaped: GPP reached a maximum around midday and only VPD, used as a downregulation function, when higher than 10 hPa, can modify this pattern that is otherwise dictated by incoming radiation. This is also visible in the modeled NEE, by comparing NNC‐part and DT outputs with measured data  TA shows a role in spring and then a less strong effect possibly due to acclimation  C‐part method uses soil temperature (in addition to air temperature) as a direct driver of RECO and the analysis of the functional responses highlighted an important and direct effect  dynamic  estimated using moving windows to indirectly consider the slow changes in water availability . The DT and NT methods use a dynamic parameter  as input in order to indirectly take into consideration footprint variability. Wind‐related variables appeared to be important in a few study sites  even though differences were detected site by site (site‐specific differences ranged between −169 and +122 g C m−2 year−1). In summary, the implemented machine learning approach gives the possibility to use variables as drivers which would be difficult to consider in process‐based approaches, which require that the role and effects of each single variable being known and correctly encoded .Finally, the standard methods do not account for EC footprint variability, which can significantly affect the magnitude of the measured fluxes if the fetch is small and not fully homogeneous (see section S8). This could be considered, for example, by fitting the models per wind sectors and wind speed classes (or using footprint models), but the amount of available data could become critically low in less represented conditions. In the NN4.3C‐part, the weights of the ANN are optimized against NEE; thus, the uncertainty of NEE could have a significant impact on their estimation. This could be particularly relevant for nighttime NEE measurements that can be affected by higher uncertainty  that, even if filtered, lead to long gaps. In these situations, a model based on ecological responses is in general more robust, and needs less data for a proper calibration than a data‐driven approach.Despite the encouraging results, there are also limitations and uncertainty sources to consider when a pure machine learning approach is used. The absence of prescribed relationships and the few constraints fixed made the method robust against incorrect or incomplete formulations used in the standard methods but also led to higher sensitivity to the uncertainty in the data used in the training phase of machine learning, particularly when both NEE and the meteorological variables used as drivers are affected by long gaps. In the NNAnother limitation is related to the availability of the large set of input data that this approach requires in order to fully exploit its advantages, which are often not fully available at all FLUXNET sites. In this situation, the model can be used with a reduced set of drivers, but this could also lead to higher uncertainty and lower performance in specific sites, while models based on ecological processes like the NT and DT may be more robust because the response pattern is already prescribed.5C‐part) using an ANN approach with a tailored structure to simultaneously retrieve GPP and RECO. From a methodological point of view, this approach is an example of simple hybrid modeling  can be exploited in future studies, for example, using as input other variables such as diffuse radiation, COS, 13C isotopes, or SIF. Additionally, NNC‐part does not make use of prescribed relationships between inputs and outputs and only a few constraints are introduced. Results in terms of the consistency of the gross CO2 estimated fluxes by NNC‐part with the estimates from FLUXNET standard methods were good and the proposed model is able to reproduce magnitudes, patterns, and functional responses. However, the systematic differences emerged by the cross‐consistency analysis and the ecological patterns captured by NNC‐part suggest that the implemented ANN reproduces fluxes dynamics more realistic than standard methods of partitioning. Being that said, further investigation are required in order to clarify the impact of the additional drivers used in NNC‐part, or possible missing functions and relationships in the standard methods.The NNAltogether, this new method provides GPP and RECO estimates that are based on purely data‐driven empirical relationships without any assumption of the driver–output relations. This product would be of high interest for process model parameterization and validation as it avoids unwanted circularity, and for this reason, we propose it as a complement to the standard processing in large networks such ICOS, AmeriFlux, or FLUXNET.Data S1Click here for additional data file.