arXiv:2605.15254cs.LG2026-05

通过空间关联设计课程学习,提升物理信息神经网络求解精度与稳定性。

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

论文配图:Curriculum Learning of Physics-Informed Neural Networks based on Spatial Correlation
图 1 · 摘自论文原文
  • 基于空间因果权重引导信息从边界向内传播,减少优化失败。
  • 引入低频信息桥抑制全局低频漂移,保持区域间一致性。
  • 自适应重加权策略恢复高频细节,适合强空间耦合的边界问题。

物理信息神经网络(PINNs)结合深度学习与物理约束,用于求解偏微分方程(PDE),广泛应用于流体力学、传热和固体力学。然而,其训练仍面临高维非凸损失景观、多目标约束不平衡及信息传播效率低等问题。现有课程学习与因果引导策略多关注时间或参数演化,缺乏对空间信息传播与区域间一致性的显式处理,且不适用于强空间耦合的边值问题(BVP)。为此,本文提出首个从子区域空间耦合角度出发的课程学习框架。首先,空间因果权重引导近边界信息向内传播,降低优化失败与虚假收敛;其次,低频信息桥通过伪标签实现空间分离区域的一致性约束,抑制全局低频漂移;第三,区域自适应重加权策略调整子区域损失,减小局部残差并恢复高频细节。在多个PDE基准测试中,该方法在相近计算成本下显著缓解训练失败,提升求解精度。代码已公开于 https://github.com/pigofmomo/CurriculumLearningPINN。

原文摘要 · Abstract (English)

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.

PINN课程学习空间耦合PDE求解

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