Physics-Informed Neural Networks and Radial Basis Functions for PDEs with Dirac Delta Sources

2026-06-10

AI Summary

This work addresses the challenge of solving partial differential equations with Dirac delta source terms using physics-informed neural networks (PINNs), which typically require smoothing approximations of the delta function and thereby introduce modeling errors. By interpreting PINNs as a residual least-squares (RLS) method, the authors propose a novel approach that directly handles the delta source through weak-form integration. They introduce a single-layer radial basis function network (RBF-RLS) tailored for transport problems. Theoretical analysis, grounded in the neural tangent kernel framework, elucidates why RBF-RLS outperforms conventional PINNs. Numerical experiments demonstrate that the method is stable and effective in both forward and inverse tasks for groundwater and riverine transport models, achieving high accuracy on synthetic, noisy, and real-world observational data.

Abstract

Physics-Informed Neural Networks (PINNs) are a machine learning method for solving forward and inverse Partial Differential Equations (PDEs). When applied to PDEs with Dirac delta functions in the forcing terms, boundary conditions, or initial conditions, PINNs require approximating them with smooth surrogate functions, a practice that can introduce significant modeling errors. In this work, we exploit the interpretation of PINNs as Residual Least Squares (RLS) methods and show that this perspective enables direct treatment of Dirac delta terms by integrating the weak-form equation. Among RLS formulations other than PINN, we focus on the Radial Basis Function (RBF) expansion (also known as a single-layer RBF Network). We show that while integrating out the Dirac delta in PINNs causes residuals to fail to converge to zero, RBF-RLS consistently provides good forward and inverse solutions to transport problems. We explain this finding using the Neural Tangent Kernel (NTK) theory. We test both approaches on linear PDEs that represent groundwater flow and transport in porous media and rivers. We solve inverse problems to fit synthetic data, noisy synthetic data, and real-world measurements.

Problem

• Physics-Informed Neural Networks
• Dirac Delta
• Partial Differential Equations
• Radial Basis Functions
• Inverse Problems

Innovation

• Physics-Informed Neural Networks
• Radial Basis Functions
• Dirac Delta
• Weak-form Integration
• Neural Tangent Kernel