Physics-Informed Machine Learning

Machine learning (ML) applications in soil science have significantly increased over the past two decades, reflecting a growing trend towards data-driven research addressing soil security. This extensive application has mainly focused on enhancing predictions of soil properties, particularly soil organic carbon, and improving the accuracy of digital soil mapping (DSM). Despite these advancements, the application of ML in soil science faces challenges related to data scarcity and the interpretability of ML models. There is a need for a shift towards Soil Science-Informed ML (SoilML) models that use the power of ML but also incorporate soil science knowledge in the training process to make predictions more reliable and generalisable. This paper proposes methodologies for embedding ML models with soil science knowledge to overcome current limitations. Incorporating soil science knowledge into ML models involves using observational priors to enhance training datasets, designing model structures which reflect soil science principles, and supervising model training with soil science-informed loss functions. The informed loss functions include observational constraints, coherency rules such as regularisation to avoid overfitting, and prior or soil-knowledge constraints that incorporate existing information about the parameters or outputs. By way of illustration, we present examples from four fields: digital soil mapping, soil spectroscopy, pedotransfer functions, and dynamic soil property models. We discuss the potential to integrate process-based models for improved prediction, the use of physics-informed neural networks, limitations, and the issue of overparametrisation. These approaches improve the relevance of ML predictions in soil science and enhance the models’ ability to generalise across different scenarios while maintaining soil science principles, transparency and reliability.

Keywords: Artificial Intelligence, Process-based models, Physics Informed Neural Networks, Informed Machine Learning, Mechanistic models, Pedology

Abstract The constitutive relations of the Richardson-Richards equation encode the macroscopic properties of soil water retention and conductivity. These soil hydraulic functions are commonly represented by models with a handful of parameters. The limited degrees of freedom of such soil hydraulic models constrain our ability to extract soil hydraulic properties from soil moisture data via inverse modeling. We present a new free-form approach to learning the constitutive relations using physically constrained neural networks. We implemented the inverse modeling framework in a differentiable modeling framework, JAX, to ensure scalability and extensibility. For efficient gradient computations, we implemented implicit differentiation through a nonlinear solver for the Richardson-Richards equation. We tested the framework against synthetic noisy data and demonstrated its robustness against varying magnitudes of noise and degrees of freedom of the neural networks. We applied the framework to soil moisture data from an upward infiltration experiment and demonstrated that the neural network-based approach was better fitted to the experimental data than a parametric model and that the framework can learn the constitutive relations.

Keywords: inverse modeling, soil hydraulic functions, physics-informed machine learning, neural networks, soil moisture

To minimize uncertainty related to soil processes in extreme events, we need accurate soil hydraulic properties across the entire range of soil water content. However, conventional methods are time-consuming and limited to specific ranges. To estimate soil hydraulic properties throughout the entire range, we conducted inverse modeling using upward infiltration experiments, where a shortwave infrared imaging camera was used to obtain high-resolution soil moisture data in space and time. Because the commonly used van Genuchten–Mualemmodel is unsuitable for describing soil hydraulic properties for dry conditions, we tested an alternative model, the Peters-Durner-Iden model, which considers both capillary and film water. The inverse modeling successfully estimated soil hydraulic properties for sandy loam and loam soils, and we demonstrated that the Peters-Durner-Iden model captured soil moisture dynamics better than the van Genuchten–Mualemmodel for dry conditions. However, both models could not adequately describe the soil moisture data for the other soils. The direct observation of the water flow via shortwave infrared images clarified that the reduced success was because of violating the one-dimensional flow assumption for coarse-textured soils and the micro-heterogeneity in soil hydraulic properties for soils with fine silt and clay materials.

Keywords: Inverse modeling, Shortwave infrared imaging, Soil moisture, Soil hydraulic functions

Water retention curve (WRC) and hydraulic conductivity function (HCF) are essential information to model the movement of water in the soil using the Richardson-Richards equation (RRE). Although laboratory measurement methods of WRC and HCF have been well established, the lab-based WRC and HCF can not be used to model soil moisture dynamics in the field because of the scale mismatch. Therefore, it is necessary to derive the inverse solution of the RRE and estimate WRC and HCF from field measurement data. We are proposing a physics-informed neural networks (PINNs) framework to obtain the inverse solution of the RRE and estimate WRC and HCF from only volumetric water content measurements. The PINNs was constructed using three feedforward neural networks, two of which were constrained to be monotonic functions to reflect the monotonicity of WRC and HCF. The PINNs was trained using noisy synthetic volumetric water content data derived from the simulation of soil moisture dynamics for three soils with distinct textures. The PINNs could reconstruct the true soil moisture dynamics from the noisy data. As for WRC, the PINN could not precisely determine the WRCs. However, it was shown that the PINNs could estimate the HCFs from only the noisy volumetric water content data without specifying initial and boundary conditions and assuming any information about the HCF (e.g., saturated hydraulic conductivity). Additionally, we showed that the PINNs framework could be used to estimate soil water flux density with a broader range of estimation than the currently available methods.

Keywords: inverse method, machine learning , partial differential equation,physics‐informed neural networks,soil moisture,soil water flux density

Water retention curve (WRC) and hydraulic conductivity function (HCF) are essential information to model the movement of water in the soil using the Richardson-Richards equation (RRE). Although laboratory measurement methods of WRC and HCF have been well established, the lab-based WRC and HCF can not be used to model soil moisture dynamics in the field because of the scale mismatch. Therefore, it is necessary to derive the inverse solution of the RRE and estimate WRC and HCF from field measurement data. We are proposing a physics-informed neural networks (PINNs) framework to obtain the inverse solution of the RRE and estimate WRC and HCF from only volumetric water content measurements. The PINNs was constructed using three feedforward neural networks, two of which were constrained to be monotonic functions to reflect the monotonicity of WRC and HCF. The PINNs was trained using noisy synthetic volumetric water content data derived from the simulation of soil moisture dynamics for three soils with distinct textures. The PINNs could reconstruct the true soil moisture dynamics from the noisy data. As for WRC, the PINN could not precisely determine the WRCs. However, it was shown that the PINNs could estimate the HCFs from only the noisy volumetric water content data without specifying initial and boundary conditions and assuming any information about the HCF (e.g., saturated hydraulic conductivity). Additionally, we showed that the PINNs framework could be used to estimate soil water flux density with a broader range of estimation than the currently available methods.

Keywords: soil water repellency, hydrophobicity ,contact angle, sorptivity, infiltration, post‐fire runoff

Understanding which soil properties control water retention at different moisture states remains a critical challenge for predicting how soils respond to management and environmental change. Traditional pedotransfer functions predict parameters of equations like van Genuchten’s model, but parameter correlations create ill-posed inverse problems that propagate errors and reduce prediction accuracy. Fundamentally, these parameter-based approaches cannot reveal which properties matter most at different water potentials or how texture-structure interactions vary across the moisture spectrum—knowledge essential for mechanistic understanding and management applications. We present Hierarchical Attention-Based Pedotransfer Function (HABIT) to address these limitations through two objectives: (1) achieve superior prediction accuracy through direct moisture prediction that circumvents parameter correlation errors, and (2) quantify property importance and interactions across moisture regimes through interpretable attention mechanisms. The model employs neural network attention architectures that dynamically weight soil properties based on water potential, enforces physical constraints including monotonicity, and accommodates incomplete soil characterization through hierarchical training. We evaluate HABIT against traditional parameter-based approaches using comprehensive international soil databases spanning global textural and structural variability. Attention weight analysis reveals moisture-dependent property interactions, including asymmetric texture-structure information flow, organic carbon’s dual structural and sorptive roles, and saturated hydraulic conductivity’s diagnostic function in integrating pore network information. These patterns generate testable predictions while explaining how direct prediction with adaptive property weighting captures soil-specific retention behaviors that parametric forms miss. HABIT demonstrates that properly designed machine learning architectures serve as prediction tools and scientific discovery platforms, providing frameworks for understanding texture-structure interactions that remain qualitatively understood but quantitatively elusive.