Curriculum Learning of Physics-Informed Neural Networks based on Spatial Correlation

TL;DR

This paper introduces a novel spatially correlated curriculum learning framework for Physics-Informed Neural Networks (PINNs) that enhances training stability and accuracy by explicitly modeling spatial information propagation and inter-region consistency.

Contribution

It is the first to address PINN training challenges through spatial coupling strategies, including causal weights, low-frequency information bridging, and adaptive loss reweighting.

Findings

Improves training stability and reduces failures in PINNs.

Enhances solution accuracy on PDE benchmarks.

Reduces global low-frequency drift during training.

Abstract

Physics-Informed Neural Networks (PINNs) combine deep learning with physical constraints for solving partial differential equations (PDEs), and are widely applied in fluid mechanics, heat transfer, and solid mechanics. However, PINN training still suffers from high-dimensional non-convex loss landscapes, imbalanced multiobjective constraints, and ineffective information propagation. Existing curriculum learning and causality-guided strategies improve training stability, but mainly focus on temporal or parametric progression, lacking explicit treatment of spatial information propagation and inter-region consistency. Moreover, they are not directly applicable to boundary value problems (BVPs) with strong spatial coupling. To address this issue, we propose a spatially correlated curriculum learning framework for PINNs. To the best of our knowledge, this is the first work to address PINN training difficulties from the perspective of spatial coupling among subregions. First, spatial causal weights guide information from near-boundary regions inward, reducing optimization failures and spurious convergence. Second, a low-frequency information bridge enforces pseudo-label-based consistency across spatially separated regions, suppressing global low-frequency drift. Third, a region-adaptive reweighting strategy adjusts subregion losses to reduce local residuals and recover high-frequency details. Experiments on PDE benchmarks show that, under comparable computational cost, the proposed method alleviates training failures and improves solution accuracy. The code is available at https://github.com/pigofmomo/CurriculumLearningPINN.