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

2026-06-10 · Source: Machine Learning · Field: Science & Research — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences, Environmental Science & Earth Systems · Depth: Expert, quick

Summary

Physics-Informed Neural Networks (PINNs) face significant modeling errors when solving Partial Differential Equations (PDEs) containing Dirac delta functions, as they typically require approximating these functions with smooth surrogates. This work reinterprets PINNs as Residual Least Squares (RLS) methods, enabling a direct treatment of Dirac delta terms through weak-form equation integration. The authors compare PINNs with Radial Basis Function (RBF) expansion, another RLS formulation. They demonstrate that while integrating out Dirac delta terms in PINNs leads to residual non-convergence, RBF-RLS consistently delivers accurate forward and inverse solutions for transport problems. This superior performance is attributed to insights from Neural Tangent Kernel (NTK) theory. The approaches were tested on linear PDEs modeling groundwater flow and river transport, solving inverse problems with synthetic, noisy synthetic, and real-world data.

Key takeaway

For research scientists developing machine learning solutions for Partial Differential Equations (PDEs) involving Dirac delta sources, you should consider Radial Basis Function (RBF)-based Residual Least Squares (RLS) methods. Standard Physics-Informed Neural Networks (PINNs) often struggle with these sources due to approximation errors and convergence issues. RBF-RLS offers a more robust approach, directly integrating weak-form equations and consistently providing accurate forward and inverse solutions, particularly for transport problems.

Key insights

RBF-RLS effectively solves PDEs with Dirac delta sources by integrating the weak-form equation, outperforming PINNs due to NTK theory.

Principles

• PINNs as RLS enable weak-form integration.
• RBF-RLS ensures residual convergence for Dirac delta.
• Smooth Dirac delta surrogates cause modeling errors.

Method

Interpret PINNs as Residual Least Squares (RLS) to integrate Dirac delta terms via weak-form equations. Compare this with Radial Basis Function (RBF)-RLS for solving forward and inverse PDEs.

In practice

• Model groundwater flow PDEs.
• Simulate river transport problems.
• Solve inverse problems with noisy data.

Topics

• Physics-Informed Neural Networks
• Radial Basis Functions
• Partial Differential Equations
• Dirac Delta Functions
• Residual Least Squares
• Neural Tangent Kernel Theory
• Inverse Problems

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.