发表机构
National University of Singapore（新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对混沌系统深度学习预测的长期误差累积问题，提出DAW框架，利用动力系统局部维度重塑损失，在KS方程上显著降低了长期自回归误差。
AI 中文摘要
用于混沌动力系统预测的深度学习替代模型在长期自回归推演中会遭遇灾难性误差累积。该行为与底层系统存在部分关联：混沌时空系统（如Kuramoto-Sivashinsky（KS）方程）在相空间的访问分布不均，以反复出现的低维静止状态（如近层流）为主，且被罕见的、动力学复杂的拓扑转变（如波合并事件）打断。在逐样本均匀目标下，标准神经替代模型会将有限容量分配给统计上数量众多的静止状态，未能充分代表会触发不成比例局部误差的瞬态区域。现有的不平衡回归方法通过目标空间密度对样本加权，但统计上的目标空间稀有性未必与内在动力学稀有性（即作为不平衡来源的吸引子的回归几何）一致。为解决该问题，我们提出动力学感知加权（Dynamics-Aware Weighting，DAW），这是一种以数据为中心的目标加权框架。利用动力系统理论中的局部维度d作为状态有效自由度的先验度量，DAW重塑损失景观，将表征能力分配给预测误差系统较大的稀疏高d区域。在混沌KS方程上，DAW始终优于均匀训练、纯统计密度加权及其随机排列的 ablation（消融实验），相对于所有基线方法降低了长期自回归误差。事件级分析表明，DAW通过抑制d急剧跳变期间产生的局部误差放大来实现这一点，而d跳变伴随KS系统中波合并等复杂物理过程。
英文摘要
Deep learning surrogates have become powerful tools for simulating and forecasting complex dynamical systems, yet their utility remains limited by catastrophic error accumulation during long-term autoregressive rollouts. This behavior is partly tied to the nature of the underlying systems: chaotic spatiotemporal systems visit phase space unevenly, with dynamics dominated by recurrent, low-dimensional quiescent states and characterized by rare and dynamically complex regime transitions. Trained under a sample-wise uniform objective, standard neural surrogates allocate their finite capacity to the statistically more numerous low-dimensional quiescent states, systematically under-representing the transient regimes that trigger disproportionate, localized errors. Existing imbalanced-regression methods tackle this issue by reweighting samples according to target-space density. However, statistical target-space rarity does not coincide with the intrinsic dynamical rarity encoded in the recurrence geometry of the attractor. To address this, we introduce Dynamics-Aware Weighting (DAW), a data-centric objective reweighting framework. Using the local dimension $d$ from dynamical systems theory as an a priori measure of a state's dynamical complexity, DAW reshapes the loss landscape to allocate representational capacity toward the sparse, high-$d$ regimes where forecast errors are systematically large. On the chaotic KS equation, DAW consistently outperforms uniform training as well as weighting based on target-space rarity, and its randomly permuted ablation, reducing long-term autoregressive error relative to all baselines. Event-level analysis shows that DAW achieves this by suppressing the localized error amplifications incurred during sharp jumps in the local dimension $d$, which typically accompany complex physical processes such as wave-merging in the KS system.
