{"ID":2891719,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.16380","arxiv_id":"2507.16380","title":"Optimization and generalization analysis for two-layer physics-informed neural networks without over-parametrization","abstract":"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 $ε\u003e0$, 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 $O(ε)$.","short_abstract":"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 trai...","url_abs":"https://arxiv.org/abs/2507.16380","url_pdf":"https://arxiv.org/pdf/2507.16380v1","authors":"[\"Zhihan Zeng\",\"Yiqi Gu\"]","published":"2025-07-22T09:24:22Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[]","has_code":false}