解决含狄拉克δ源项的偏微分方程求解难题，提升物理信息神经网络精度
Physics-Informed Neural Networks and Radial Basis Functions for PDEs with Dirac Delta Sources
- 将PINN视为弱形式残差最小化，直接处理狄拉克δ源项
- 基于径向基函数的RLS方法在输运问题中实现稳定收敛
- 适用于地下水流、河流污染物传输等实际场景的逆问题求解
物理信息神经网络（PINNs）是一种求解前向与反向偏微分方程（PDEs）的机器学习方法。当处理强迫项、边界条件或初始条件中含有狄拉克δ函数的PDE时，传统PINN需用光滑替代函数近似δ函数，这会引入显著建模误差。本文将PINN视为残差最小二乘（RLS）方法，发现该视角可通过对弱形式方程进行积分，直接处理狄拉克δ项。我们对比研究了其他非PINN类的RLS方法，重点关注径向基函数（RBF）展开（即单层RBF网络）。结果表明：尽管在PINN中积分δ项会导致残差无法收敛至零，但RBF-RLS在输运问题上始终能提供良好前向与反向解。我们借助神经正切核（NTK）理论解释该现象。在描述多孔介质中地下水流动与输运、河流系统的线性PDE上进行了测试，通过拟合合成数据、含噪合成数据及真实测量数据求解逆问题。
原文摘要 · Abstract (English)
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.
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