分阶段深度训练：PINNs 的表征课程
Staged Depth Training: A Representation Curriculum for PINNs
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中文总结 AI 辅助
本文提出表征课程方法，通过分阶段深度训练（SDT）显式学习并逐步细化表征，在不改变最终架构和推理成本的情况下，显著提升 PINNs 在多个基准问题上的性能。
中文摘要 AI 辅助
表征质量是决定 PINNs 性能的核心因素，然而标准训练在拟合最终解的同时，让表征隐式地涌现。我们引入了“表征课程”，这是一种有序的过程，其中表征被显式地学习，独立于其预测器进行迁移，并逐步细化。我们通过分阶段深度训练（SDT）实现这一过程，该方法在临时的物理信息头下训练一个浅层前缀，丢弃该头，并在增加深度的同时冻结已学习的前缀，无需特定于方程的编码或对最终架构的更改。在 PINNacle 的 20 个默认正向问题中，使用三种骨干网络，SDT 在 59 个等预算问题-骨干组合中，有 40 个至少提升了 5%，其余组合也保持在该范围内，在 PirateNet 风格骨干上实现了 32.8% 的几何平均误差降低。机制消融表明，这种提升不能仅由优化器重启或浅层热启动来解释。表征可视化和超参数盆地分析提供了关于表征几何形状和对共享超参数的局部敏感性的诊断证据。在二维 Poisson-Boltzmann 问题上，SDT 还将两种骨干的拟合深度缩放指数提高了一倍以上。这些结果支持表征课程作为一种有前景的训练策略，可在保持部署架构和推理成本的同时改进 PINNs。
英文摘要
Representation quality is a central determinant of PINNs' performance, yet standard training leaves representations to emerge implicitly while fitting the final solution. We introduce \textbf{representation curriculum}, an ordered process in which representations are explicitly learned, transferred independently of their predictors, and progressively refined. We realize it with Staged Depth Training (SDT), which trains a shallow prefix under a temporary physics-informed head, discards the head, and freezes the learned prefix while adding depth, without equation-specific encodings or changes to the final architecture. Across the 20 default forward problems in PINNacle with three backbones, SDT improves 40 of 59 equal-budget problem--backbone cells by at least 5\% and remains within that band in the rest, with a 32.8\% geometric-mean error reduction on a PirateNet-style backbone. Mechanistic ablations suggest that the gain is not explained by optimizer restarts or shallow warm-starting alone. Representation visualizations and hyperparameter-basin analyses provide diagnostic evidence on representation geometry and local sensitivity to shared hyperparameters. On Poisson--Boltzmann 2D, SDT also more than doubles the fitted depth-scaling exponent for both backbones. These results support representation curriculum as a promising training strategy for improving PINNs while preserving the deployed architecture and inference cost.
