Practical Application of Physics-Informed Neural Networks in Groundwater Flow Simulation

This article introduces an open-source project (PINNs-in-Groundwater) for hydrogeology and water resources researchers, showing how to use PyTorch to build Physics-Informed Neural Networks (PINNs) to solve 1D and 2D groundwater flow problems. Addressing the limitations of traditional numerical simulation methods, the project provides a complete workflow from automatic differentiation to boundary condition handling, including visualization and validation methods, offering valuable resources for learners new to PINNs.

The mathematical model of groundwater flow is based on partial differential equations (PDEs). Traditional numerical methods (finite element, finite difference) require discrete meshes, but face issues such as time-consuming mesh generation for complex geometries, unstable inversion when data is sparse, and high computational costs for real-time prediction. By integrating deep learning with physical laws, PINNs can learn PDE solutions without meshes, providing a new approach to address these challenges.

PINNs integrate physical laws as soft constraints into training: 1. Use deep neural networks to approximate unknown functions (e.g., groundwater head distribution); 2. Compute derivatives of all orders using PyTorch's automatic differentiation; 3. Construct a physical loss function that includes PDE residuals and the degree of violation of initial/boundary conditions; 4. Use multi-task learning to simultaneously fit observed data and satisfy physical constraints, enabling reasonable predictions even when data is scarce.

This GitHub project uses a progressive design: the basic module introduces PyTorch neural networks and automatic differentiation; the 1D problem module demonstrates PINNs applications for steady-state flow and boundary condition handling; the 2D extension module deals with complex geometries and visualization; the advanced application module explores coupled problems such as transient flow and contaminant transport.

Key technical points include: network architecture selection (fully connected networks, which need sufficient capacity to capture spatial variations); loss function balancing (weight adjustment for data loss, PDE residual loss, and boundary condition loss); training strategies (learning rate scheduling, Adam + L-BFGS optimizer combination); post-processing and visualization (plotting head distribution, streamline diagrams, and error maps).

PINNs are suitable for scenarios such as parameter inversion (estimating aquifer parameters using sparse observations), data assimilation (combining real-time monitoring with physical models), surrogate models (low-computational-cost alternatives for optimization problems), and uncertainty quantification (estimating prediction uncertainty via ensemble models or Bayesian frameworks), with broad application prospects.

PINNs represent an important direction in scientific machine learning and expand the toolbox for hydrogeological research. This open-source project has a clear structure and detailed documentation, making it a high-quality resource for getting started with PINNs. It is recommended that researchers try this project, explore advanced applications of PINNs in combination with practical problems, and promote their implementation in water resource management and environmental protection.