Adaptive residual splitting in PINNs for solving complex PDEs

When Physics Informed Neural Networks (PINNs) are used to solve complex PDEs, they often encounter accuracy problems due to the PDE residual term governing the other loss components, affecting the convergence of the overall loss function. Hence, this study proposes an adaptive residual splitting PINN (ARSPINN) that decomposes the PDE residual into multiple subterms and utilizes an adaptive loss weight strategy to adjust their weights. The developed scheme breaks down the complex global PDE residuals, mitigating the challenges of global optimization and alleviating the problem of global loss residuals masking local features. Moreover, to improve the convergence of the loss function further, a three-stage hybrid optimization strategy is introduced that leverages the advantages of various optimization algorithms in different training stages, allowing fast convergence and high-accuracy outputs for complex PINN models. Several numerical examples compare the accuracy and effectiveness of the ARSPINN method with traditional global loss term balancing and point-wise loss balancing algorithms, demonstrating that ARSPINN outperforms various task-based global loss term weight balancing techniques using PINN in terms of accuracy. Additionally, ARSPINN improves performance over point-wise residual-based weight balancing methods regarding CPU time and convergence.