Artificial Neural Networks for the Identification of Partial Differential Equations of Land Surface Schemes in Climate Models

Abstract

Land surface scheme in climate models is a solver for a nonlinear PDE system, which describes thermal conductance and water diffusion in soil. Thermal conductivity $\lambda_T$, water diffusivity $\lambda_W$ and hydraulic conductivity $\gamma$ coefficients of this system are functions of the solution of the system $W$ and $T$. For the climate models to accurately represent the Earth system's evolution, one needs to approximate the coefficients or estimate their values empirically. Measuring the coefficients is a complicated in-lab experiment without a chance to cover the full range of environmental conditions. In this work, we propose a data-driven approach for approximating the parameters of the PDE system describing the evolution of soil characteristics. We formulate the coefficients as parametric functions, namely artificial neural networks. We propose training these neural networks with the loss function computed as a discrepancy between the PDE system solution and the measured characteristics $W$ and $T$. We also propose a scheme inherited from the backpropagation method for calculating the gradients of the loss function w.r.t. network parameters. As a proof-of-concept step, we assessed the capabilities of our approach in three synthetic scenarios: a nonlinear thermal diffusion equation, a nonlinear liquid water $W$ diffusion equation, and Richards equation. We generated realistic initial conditions and simulated synthetic evolutions of $W$ and $T$ that we used as measurements in the networks` training procedure for these three scenarios. The results of our study show that our approach provides an opportunity for reconstructing the PDE coefficients of various forms accurately without actual knowledge of their ground truth values.