# Optimization and generalization analysis for two-layer physics-informed neural networks without over-parametrization

arXiv: https://arxiv.org/abs/2507.16380 · HTML: https://mlsanity.com/paper/2507.16380/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- First posted on arXiv 22 Jul 2025.

## Abstract (arXiv)

This work focuses on the behavior of stochastic gradient descent (SGD) in solving least-squares regression with physics-informed neural networks (PINNs). Past work on this topic has been based on the over-parameterization regime, whose convergence may require the network width to increase vastly with the number of training samples. So, the theory derived from over-parameterization may incur prohibitive computational costs and is far from practical experiments. We perform new optimization and generalization analysis for SGD in training two-layer PINNs, making certain assumptions about the target function to avoid over-parameterization. Given ε>0, we show that if the network width exceeds a threshold that depends only on ε and the problem, then the training loss and expected loss will decrease below O(ε).