Abstract
ABSTRACT This paper investigates the application of Physics-Informed Neural Networks (PINNs) to inverse problems in unsaturated groundwater flow. PINNs are applied to the types of unsaturated groundwater flow problems modelled with the Richards partial differential equation and the van Genuchten constitutive model. The inverse problem is formulated here as a problem with known or measured values of the solution to the Richards equation at several spatio-temporal instances, and unknown values of solution at the rest of the problem domain and unknown parameters of the van Genuchten model. PINNs solve inverse problems by reformulating the loss function of a deep neural network such that it simultaneously aims to satisfy the measured values and the unknown values at a set of collocation points distributed across the problem domain. The novelty of the paper originates from the development of PINN formulations for the Richards equation that requires training of a single neural network. The results demonstrate that PINNs are capable of efficiently solving the inverse problem with relatively accurate approximation of the solution to the Richards equation and estimates of the van Genuchten model parameters.
Highlights
Predictions of various problems in unsaturated soil mechanics, such as rainfall-induced landslides and excavation collapses, rely on thorough understanding and the capacity to model coupled hydro-mechanical process in soils (e.g. Zhang et al, 2018)
This study examines the capacity of Physics-Informed Neural Networks (PINNs) in solving unsaturated groundwater flow problems modelled by the Richards Partial Differential Equation (PDE) (Richards, 1931)
This study investigated the application of physicsinformed neural networks to the inverse problem for the Richards partial differential equation with the van Genuchten model
Summary
Predictions of various problems in unsaturated soil mechanics, such as rainfall-induced landslides and excavation collapses, rely on thorough understanding and the capacity to model coupled hydro-mechanical process in soils (e.g. Zhang et al, 2018). The nonlinear nature of this equation is a reflection of the nonlinear relationship between the soil volumetric water content, θ, and the pressure head, ψ, which together with hydraulic conductivity, k form three primary variables of Richards equation. The primary variables are mutually dependent with θ and k often being expressed as a function of ψ, respectively, with the Water Retention Curve (WRC) and Hydraulic Conductivity Function (HCF). Both of these hydraulic functions are known as constitutive relationships and are utilised to describe the characteristics of water and solute movement in soils As a result of the simulation both ψ and θ values are known at the 40,000 points across the spatio-temporal domain
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