Divide and Conquer: Region Self-Paced Physics-Informed Neural Networks

Recently, Physics-Informed Neural Networks (PINNs) have shown strong potential in scientific computing and engineering simulation. Although existing PINN methods have achieved promising progress, they still suffer from the Unbalanced Prediction Problem (UPP), where challenging regions are learned much worse than smooth regions. To alleviate this issue, adaptive sampling and cognitive learning strategies have been introduced to emphasize hard samples by dynamically evaluating sample difficulty. However, the coupling between sample selection and difficulty evaluation may make many samples receive nearly indistinguishable difficulty scores, leading to the Difficulty Cohesion Problem (DCP). This problem weakens the intended easy-to-hard curriculum and limits the learning efficiency of PINNs. To address both UPP and DCP, we propose Region Self-Paced Physics-Informed Neural Networks (RSPINNs), a divide-and-conquer framework for time-dependent PDEs. Specifically, RSPINNs partitions the temporal domain into multiple subdomains and evaluates sample difficulty independently within each subdomain using PDE residual gradients. Then, a rank-based normalization strategy preserves the relative easy-to-hard ordering and improves difficulty discrimination. Finally, a bimodal self-paced scheduler converts the ranked difficulty index into smooth sample weights, guiding the model to progressively shift attention from easy to hard samples. Extensive experiments on six PDE benchmarks demonstrate that RSPINNs consistently improves prediction accuracy in challenging regions and achieves competitive overall performance with practical computational cost.