Dynamic Curriculum Regularization for Enhanced Training of Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) are increasingly used to solve differential and integral equations, particularly nonlinear partial differential equations (PDEs), but often struggle with complex industrially-relevant problems. We introduce a new curriculum regularisation scheme for training PINNs, addressing two key limitations in existing methods. First, we implement early stopping, based on the maximum residual of the PINN solution, to prevent high-residual regions from propagating. Second, we propose a dynamic adjustment of the PDE parameter step size, guided by the $\mathcal{L}$$_2$ distance between solutions, to handle varying solution complexities in the PDE and PINN solution quality. Our approach improves both the $\mathcal{L}$$_2$ relative error and convergence speeds across three PDEs, demonstrating greater robustness and efficiency over traditional curriculum regularisation. We predict this technique to generalise well to other PDEs where vanilla PINNs fail to converge.