Statistical Learning Analysis of Physics-Informed Neural Networks

• URL: http://arxiv.org/abs/2602.11097v1
• Date: Wed, 11 Feb 2026 18:09:29 GMT
• Title: Statistical Learning Analysis of Physics-Informed Neural Networks
• Authors: David A. Barajas-Solano,
• Abstract summary: We study the training and performance of physics-informed learning for initial and boundary value problems with physics-informed neural networks (PINNs)<n>We use the so-called Local Learning Coefficient to analyze the estimates of PINN parameters obtained via optimization for a heat equation.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the training and performance of physics-informed learning for initial and boundary value problems (IBVP) with physics-informed neural networks (PINNs) from a statistical learning perspective. Specifically, we restrict ourselves to parameterizations with hard initial and boundary condition constraints and reformulate the problem of estimating PINN parameters as a statistical learning problem. From this perspective, the physics penalty on the IBVP residuals can be better understood not as a regularizing term bus as an infinite source of indirect data, and the learning process as fitting the PINN distribution of residuals $p(y \mid x, t, w) q(x, t) $ to the true data-generating distribution $δ(0) q(x, t)$ by minimizing the Kullback-Leibler divergence between the true and PINN distributions. Furthermore, this analysis show that physics-informed learning with PINNs is a singular learning problem, and we employ singular learning theory tools, namely the so-called Local Learning Coefficient (Lau et al., 2025) to analyze the estimates of PINN parameters obtained via stochastic optimization for a heat equation IBVP. Finally, we discuss implications of this analysis on the quantification of predictive uncertainty of PINNs and the extrapolation capacity of PINNs.

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