发表机构
RIKEN Center for Advanced Intelligence Project; Graduate School of Information Science and Technology, The University of Osaka（理化学研究所先进智能项目中心; 大阪大学信息科学与技术研究生院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究通过分析Koopman算子的谱可靠性，对比两种损失训练的Koopman自编码器在六个混沌系统上的滚动误差等指标，发现谱残差模型性能更优，验证了谱可靠性对潜维缩放的解释作用。
AI 中文摘要
在深度学习中，逼近理论促使人们增大表示规模。我们探究这种益处是否能通过自回归预测延伸至动力学学习。我们通过Koopman算子的本征结构分析学习到的时间演化，利用相对残差检测即使在单步误差下降时仍会出现的虚假本征对。对于有界Koopman算子，我们证明，当学习到的字典空间在$L^2$中逼近可观测量空间时，这些空间上的最小残差会逐点收敛到全空间对应值。我们的假设是，Koopman谱可靠性有助于解释滚动误差是否随维度增加而持续下降。我们比较了两种共享Koopman自编码器的模型，该自编码器交替针对重构和潜演化进行训练，使用潜预测损失（潜坐标中的单步预测误差）或谱残差损失（候选本征对的相对残差）。在六个混沌系统上，两种模型均将中位数窗口滚动误差从最小维度降至最大维度。谱残差模型在所有系统和维度上的中位数均低于潜预测模型，且在每个系统中其中位数下降幅度更大。在四个系统中，其中位数随维度单调下降，而潜预测模型仅在一个系统中如此。与四个基准族相比，其平均有效预测时间几乎总是更长。在两阶段训练的最大维度下，我们将本征值位置与每个学习到的字典的残差轮廓进行比较。谱残差本征值集中在低残差区域，而潜预测本征值也出现在高残差区域，这与我们的假设一致。
英文摘要
In deep learning, approximation theory motivates increasing representation size. We ask whether this benefit extends to dynamics learning through autoregressive prediction. We analyze the learned time evolution through the eigenstructure of Koopman operators, using relative residuals to detect spurious eigenpairs arising even as one-step error falls. For bounded Koopman operators, we show that minimal residuals over learned dictionary spaces converge pointwise to their full-space counterparts as these spaces approximate the observable space in $L^2$. Our hypothesis is that Koopman spectral reliability helps explain how consistently rollout error decreases with increasing dimension. We compare two models of a shared Koopman autoencoder trained alternately for reconstruction and latent evolution, using latent-prediction loss (one-step prediction errors in latent coordinates) or spectral-residual loss (relative residuals of candidate eigenpairs). Across six chaotic systems, both models reduced median windowed rollout error from smallest to largest dimension. The spectral-residual model achieved lower medians than the latent-prediction model for all systems and dimensions, and its median fell by a larger factor in every system. Its median decreased monotonically with dimension in four systems, against one for latent prediction. Against four baseline families, its mean valid prediction times were nearly always longer. At the largest dimension under two-stage training, we compared eigenvalue positions with each learned dictionary's residual contours. Spectral-residual eigenvalues concentrated in low-residual regions, whereas latent-prediction eigenvalues also appeared in high-residual regions, consistent with the hypothesis.
