# Statistical Learning Analysis of Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2602.11097 · HTML: https://mlsanity.com/paper/2602.11097/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 16 references that are in the map.
- First posted on arXiv 11 Feb 2026.

## Builds on (most cited first, 7 of 7)

- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [The Local Learning Coefficient: A Singularity-Aware Complexity Measure](https://mlsanity.com/paper/2308.12108/) (2023)
- [Loss Landscape Degeneracy and Stagewise Development in Transformers](https://mlsanity.com/paper/2402.02364/) (2024)
- [BlackJAX: Composable Bayesian inference in JAX](https://mlsanity.com/paper/2402.10797/) (2024)
- [Estimating Real Log Canonical Thresholds](https://mlsanity.com/paper/1906.01341/) (2019)
- [Randomized Physics-Informed Neural Networks for Bayesian Data Assimilation](https://mlsanity.com/paper/2407.04617/) (2024)

## Built on by (most cited first, 1 of 1)

- [When Known Physics Helps Neural PDE Models: Residual Constraints Out-Regularize Generic Priors for Nonlinear Dynamics](https://mlsanity.com/paper/2609.34012/) (2026)

## Abstract (arXiv)

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.