Parameterised PINNs for water infiltration in unsaturated soils

Modelling water infiltration in unsaturated soils is vital for maintaining a healthy ecosystem, analysing the stability of slopes, or promoting sustainable agriculture. Recently, Physics-informed Neural Networks (PINNs) have gained popularity in solving highly nonlinear problems like the Richardson-Richards equation (RRE), by approximating physical laws with a loss term in a mesh-free approach, often using sparse data points, to mimic the gap spacing between field sensors. However, despite several successful applications in modelling 1D infiltration problems, the generalisation capability of these models is often limited by the specific scenarios used during training. Therefore, potential of the neural networks as universal approximators are not exploited in such applications. This paper investigates the feasibility of applying a Parameterised-PINNs (P-PINNs) as a surrogate model to solve the RRE. The model was trained only once across a range of infiltration conditions defined by varying soil hydraulic properties and meteorological conditions to evaluate its ability to predict various scenarios within the multidimensional parameter space without additional observation data. Results show that a wider rather than a deeper network architecture, enhanced by dynamic adaptive techniques, such as time-stratified Residual-based Adaptive Refinement (RAR), Layer-wise Locally Adaptive Activation Function (L-LAAF), and Principled Loss Function (PLF), aids in capturing the correct physical profile. Although the model achieved high overall performance when validated against analytical solutions, Nash-Sutcliffe Efficiency (NSE) > 0.99, it exhibited very minor phase errors. P-PINN was tested across drastically changing parameters, e.g. soils with very high or very low air-entry values, and satisfactory validation metrics were obtained. Our implementation P-PINNs demonstrate the potential as a universal non-linear approximator for such problems, where the initial computational cost of training is offset by the instant large-scale evaluations.