Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspective
This work addresses accuracy issues in PINNs for solving PDEs, but it appears incremental as it builds on existing methods with a new weighting approach.
The paper tackles the challenge of convergence and accuracy in physics-informed neural networks (PINNs) when optimized with finite point sets, proposing an adaptive weighting scheme that reduces relative L2 errors.
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.