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.

翻译：物理信息神经网络（PINNs）是一种用于求解正向和逆向偏微分方程（PDEs）的机器学习方法。当应用于强迫项、边界条件或初始条件中包含狄拉克δ函数的偏微分方程时，PINNs需要采用光滑代理函数近似这些奇异项，这种做法可能引入显著建模误差。本文利用PINNs作为残差最小二乘法（RLS）的数学诠释，证明该视角可通过积分弱形式方程直接处理狄拉克δ项。在PINN之外的RLS公式中，我们重点关注径向基函数（RBF）展开（亦称单层RBF网络）。研究表明，PINNs中积分消除狄拉克δ会导致残差无法收敛至零，而RBF-RLS方法能持续为输运问题提供高质量的正向和逆向解。我们通过神经正切核（NTK）理论解释这一现象。我们在代表多孔介质与河流中地下水流动及输运的线性偏微分方程上测试了两种方法，并通过求解逆向问题拟合合成数据、含噪合成数据及实测数据。