arXiv:2608.22277cs.LGphysics.comp-ph2026-08

针对混沌系统预测中误差累积问题,提出动态感知加权方法提升高复杂度状态的建模精度。

DAW: Dynamics-Aware Weighting for Deep Learning Forecasts of Chaotic Systems

论文配图:DAW: Dynamics-Aware Weighting for Deep Learning Forecasts of Chaotic Systems
图 1 · 摘自论文原文
  • 基于局部维度$d$构建动态感知加权框架,优先分配模型容量给高复杂度状态
  • 在KS方程上长期预测误差显著降低,尤其抑制了$ d $突增时的局部误差放大
  • 适合需要精准捕捉突发物理过程(如波合并)的混沌系统建模任务

深度学习替代模型在长期自回归滚动预测混沌动力系统时存在灾难性误差累积。这与系统内在特性相关:如柯朗-希瓦辛斯基(KS)方程这类时空混沌系统在相空间分布不均——以频繁出现的低维静止态为主,偶发稀疏但动态复杂的拓扑跃迁(如波合并事件)。标准神经网络在样本均匀目标下,将有限容量偏向统计上常见的静止态,忽视引发显著局部误差的瞬态过程。现有不平衡回归方法通过目标空间密度重加权,但统计稀有性未必对应动力学稀有性。为此提出动态感知加权(DAW),利用动力系统理论中的局部维度 $d$ 作为状态活跃自由度的先验度量,重构损失函数,使模型容量向稀疏、高 $d$ 的区域倾斜,从而提升对大误差区域的建模能力。在KS方程上,DAW持续优于均匀训练、纯统计密度加权及其随机置换基线,显著降低长期自回归误差。事件级分析表明,其通过抑制 $d$ 突升时产生的局部误差放大,有效应对波合并等复杂物理过程。

原文摘要 · Abstract (English)

Deep learning surrogates for forecasting chaotic dynamical systems suffer from catastrophic error accumulation over long-term autoregressive rollouts. This behavior is partly tied to the underlying systems: chaotic spatiotemporal systems, such as the Kuramoto-Sivashinsky (KS) equation, visit phase space unevenly - dominated by recurrent, low-dimensional quiescent states (e.g., near-laminar flows) and punctuated by rare, dynamically complex topological transitions (e.g., wave-merging events). Under a sample-wise uniform objective, standard neural surrogates allocate their finite capacity to the statistically numerous quiescent states, under-representing the transient regimes that trigger disproportionate, localized errors. Existing imbalanced-regression methods reweight samples by target-space density. However, statistical target-space rarity need not coincide with the intrinsic dynamical rarity - the recurrence geometry of the attractor that is the source of the imbalance. 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 active degrees of freedom, 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, purely statistical density weighting, 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 $d$, which accompany complex physical processes such as wave-merging in the KS system.

混沌系统动态加权误差控制深度学习

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