Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspective

• URL: http://arxiv.org/abs/2506.19805v2
• Date: Fri, 18 Jul 2025 09:09:51 GMT
• Title: Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspective
• Authors: Chenhao Si, Ming Yan,
• Abstract summary: Physics-informed neural networks (PINNs) are extensively employed to solve partial differential equations (PDEs)<n>PINNs are typically optimized using a finite set of points, which poses significant challenges in guaranteeing their convergence and accuracy.<n>We propose a new weighting scheme that will adaptively change the weights to the loss functions from isolated points to their continuous neighborhood regions.
• Score: 14.65008276932511
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Physics-informed neural networks (PINNs) are extensively employed to solve partial differential equations (PDEs) by ensuring that the outputs and gradients of deep learning models adhere to the governing equations. However, constrained by computational limitations, PINNs are typically optimized using a finite set of points, which poses significant challenges in guaranteeing their convergence and accuracy. In this study, we proposed a new weighting scheme that will adaptively change the weights to the loss functions from isolated points to their continuous neighborhood regions. The empirical results show that our weighting scheme can reduce the relative $L^2$ errors to a lower value.

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