改进降维模型的潜在空间几何结构，提升长期预测性能。
On Geometry Regularization in Autoencoder Reduced-Order Models with Latent Neural ODE Dynamics
- 用神经微分方程建模潜空间动态，结合四类几何正则化方法。
- 仅首层斯特凡投影能稳定潜空间，改善长时序滚动预测效果。
- 潜空间几何失配会损害下游动态建模，超越局部平滑收益。
本文研究编码器-解码器降维模型中学习到的潜空间表示的几何正则化策略。在对对流-扩散-反应（ADR）方程的固定实验设置下，采用神经微分方程建模潜空间动态，并评估四种在自编码器预训练阶段应用的正则化方法：(a) 解码器雅可比矩阵的近等距正则化，(b) 基于随机方向增益的随机解码器增益惩罚，(c) 二阶方向曲率惩罚，以及 (d) 解码器第一层的斯特凡（Stiefel）投影。在多个随机种子下，发现 (a)–(c) 方法虽提升解码器局部平滑性或敏感度代理指标，但常使后续使用冻结自编码器进行潜空间动力学训练更困难，尤其在长时序滚动预测中表现不佳。相比之下，(d) 方法始终改善了所学潜空间动力学的条件相关诊断指标，并倾向于获得更好的滚动预测性能。我们提出假设：在此设定下，潜空间几何不匹配的下游影响超过了解码器平滑性的收益。
原文摘要 · Abstract (English)
We investigate geometric regularization strategies for learned latent representations in encoder--decoder reduced-order models. In a fixed experimental setting for the advection--diffusion--reaction (ADR) equation, we model latent dynamics using a neural ODE and evaluate four regularization approaches applied during autoencoder pre-training: (a) near-isometry regularization of the decoder Jacobian, (b) a stochastic decoder gain penalty based on random directional gains, (c) a second-order directional curvature penalty, and (d) Stiefel projection of the first decoder layer. Across multiple seeds, we find that (a)--(c) often produce latent representations that make subsequent latent-dynamics training with a frozen autoencoder more difficult, especially for long-horizon rollouts, even when they improve local decoder smoothness or related sensitivity proxies. In contrast, (d) consistently improves conditioning-related diagnostics of the learned latent dynamics and tends to yield better rollout performance. We discuss the hypothesis that, in this setting, the downstream impact of latent-geometry mismatch outweighs the benefits of improved decoder smoothness.
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