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

• URL: http://arxiv.org/abs/2507.16380v1
• Date: Tue, 22 Jul 2025 09:24:22 GMT
• Title: Optimization and generalization analysis for two-layer physics-informed neural networks without over-parametrization
• Authors: Zhihan Zeng, Yiqi Gu,
• Abstract summary: This work focuses on the behavior of gradient descent (SGD) in solving least-squares regression with physics-informed neural networks (PINNs)<n>We show that if the network width exceeds a threshold that depends only on $epsilon$ and the problem, then the training loss and expected loss will decrease below $O(epsilon)$.
• Score: 0.6215404942415159
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $\epsilon>0$, we show that if the network width exceeds a threshold that depends only on $\epsilon$ and the problem, then the training loss and expected loss will decrease below $O(\epsilon)$.

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