Physics-Informed Machine Learning for Modeling Unsaturated Soil Water Flow
(2026) In Master's Theses in Mathematical Sciences BERM02 20261Mathematics (Faculty of Sciences)
- Abstract
- Unsaturated soil water flow is commonly described by Richards’ equation, a nonlinear PDE that is typically solved using numerical methods. In this thesis, two alternative physics-informed machine learning approaches are investigated for forward and inverse modeling of unsaturated flow. First, physics-informed neural networks (PINNs) are applied to approximate solutions of Richards’ equation in a one-dimensional homogeneous soil setting. Their performance is evaluated by comparison with numerical reference solutions. In addition, an inverse PINN
framework is employed to estimate the soil hydraulic parameters α, n and the saturated hydraulic conductivity Ks of the van Genuchten model. Second, a physics-informed deep operator network... (More) - Unsaturated soil water flow is commonly described by Richards’ equation, a nonlinear PDE that is typically solved using numerical methods. In this thesis, two alternative physics-informed machine learning approaches are investigated for forward and inverse modeling of unsaturated flow. First, physics-informed neural networks (PINNs) are applied to approximate solutions of Richards’ equation in a one-dimensional homogeneous soil setting. Their performance is evaluated by comparison with numerical reference solutions. In addition, an inverse PINN
framework is employed to estimate the soil hydraulic parameters α, n and the saturated hydraulic conductivity Ks of the van Genuchten model. Second, a physics-informed deep operator network (DeepONet) is employed to learn the mapping between varying upper boundary flux functions and the corresponding solutions of Richards’ equation. In this context, the model’s ability to generalize across different boundary conditions is assessed. (Less) - Popular Abstract
- Water flow through soil is an important process in many environmental applications. In agriculture, for example, understanding how water infiltrates and moves through the ground is essential for efficient irrigation and sustainable water management. These processes can be described mathematically by using differential equations that represent the physical laws driving water movement in soil. One of the most important equations for this purpose is the so-called Richards' equation, a complex nonlinear partial differential equation. Solving this equation typically requires numerical methods that approximate the solution step by step on a spatial and temporal grid. While these methods can produce accurate results, they often require... (More)
- Water flow through soil is an important process in many environmental applications. In agriculture, for example, understanding how water infiltrates and moves through the ground is essential for efficient irrigation and sustainable water management. These processes can be described mathematically by using differential equations that represent the physical laws driving water movement in soil. One of the most important equations for this purpose is the so-called Richards' equation, a complex nonlinear partial differential equation. Solving this equation typically requires numerical methods that approximate the solution step by step on a spatial and temporal grid. While these methods can produce accurate results, they often require considerable computational effort, especially for large or complex problems.
An alternative approach is to use machine learning methods, such as artificial neural networks. Instead of solving the equation numerically, a neural network can learn to approximate its solution based on the known physical laws, i.e., the governing differential equation of the system. In this context, the term physics-informed machine learning is used because the neural network does not learn only from data but also from equations that describe the underlying physical processes. In this thesis, two different physics-informed machine learning approaches are investigated.
First, physics-informed neural networks (PINNs) are used to approximate solutions of Richards’ equation for a simplified soil water flow problem. The obtained predictions are then compared with classical numerical solutions generated with a standard hydrological simulation model. In addition, PINNs are applied in an inverse modeling setup, where the goal is to estimate important soil hydraulic parameters that are part of the mathematical model. Secondly, deep operator networks (DeepONets) are investigated as a more general operator-learning approach. In this method, the neural network learns relationships between input functions and the corresponding solution functions. Unlike PINNs, which solve one specific problem setup, DeepONets aim to generalize across varying boundary conditions and approximate the corresponding solutions of the equation. This enables predictions for previously unseen infiltration scenarios, for example, due to changing rainfall patterns. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/student-papers/record/9244270
- author
- Grimm, Jule LU
- supervisor
- organization
- course
- BERM02 20261
- year
- 2026
- type
- H2 - Master's Degree (Two Years)
- subject
- publication/series
- Master's Theses in Mathematical Sciences
- report number
- LUNFBV-3009-2026
- ISSN
- 1404-6342
- other publication id
- 2026:E50
- language
- English
- id
- 9244270
- date added to LUP
- 2026-09-14 16:21:38
- date last changed
- 2026-09-14 16:21:38
@misc{9244270,
abstract = {{Unsaturated soil water flow is commonly described by Richards’ equation, a nonlinear PDE that is typically solved using numerical methods. In this thesis, two alternative physics-informed machine learning approaches are investigated for forward and inverse modeling of unsaturated flow. First, physics-informed neural networks (PINNs) are applied to approximate solutions of Richards’ equation in a one-dimensional homogeneous soil setting. Their performance is evaluated by comparison with numerical reference solutions. In addition, an inverse PINN
framework is employed to estimate the soil hydraulic parameters α, n and the saturated hydraulic conductivity Ks of the van Genuchten model. Second, a physics-informed deep operator network (DeepONet) is employed to learn the mapping between varying upper boundary flux functions and the corresponding solutions of Richards’ equation. In this context, the model’s ability to generalize across different boundary conditions is assessed.}},
author = {{Grimm, Jule}},
issn = {{1404-6342}},
language = {{eng}},
note = {{Student Paper}},
series = {{Master's Theses in Mathematical Sciences}},
title = {{Physics-Informed Machine Learning for Modeling Unsaturated Soil Water Flow}},
year = {{2026}},
}