A residual weighted physics informed neural network for forward and inverse problems of reaction diffusion equations

Abstract

Physics-Informed Neural Networks (PINNs) constitute an effective deep learning approach for solving partial differential equations (PDEs). In this work, we propose a Residual-Weighted Physics-Informed Neural Network (RW-PINN), a new method designed to enhance the accuracy of PINNs -based algorithms. We construct a deep learning framework with two residual-weighting schemes to solve reaction–diffusion equations and evaluate its performance on both forward and inverse problems. The approach computes weights proportional to the PDE residuals, rescales them, and incorporates these scaled residuals into the loss function, leading to more stable training. Numerical experiments on nonlinear equations, complemented by sensitivity analysis and statistical error analysis, are conducted to validate the proposed algorithms. To further demonstrate the effectiveness of our methodology, we implemented PINN-based forward (Mishra and Molinaro, 2023; Bai et al., 2021) and inverse problems (Mishra and Molinaro, 2022) for the nonlinear equations and conducted a comparative analysis with the proposed RW-PINN approach.