一族Kuramoto-Sivashinsky偏微分方程不变测度的摊销生成建模

中文摘要 AI 辅助

混沌耗散偏微分方程的长期统计由不变测度描述，当方程携带参数时，这些测度构成一个族，其成员在类型上可能不同，从稳态到持续时空混沌。我们提出一个问题：一个单一的生成模型能否覆盖这样的族。我们仅训练一次单一的条件扩散模型，针对Kuramoto-Sivashinsky方程在160个超粘性参数值下的状态，并使用分类器双样本检验（辅以谱、几何和尾部统计）在训练中未见过的交错参数值上对保留数据进行评估。在该族的大部分成员上，随着动力学变得更加混沌，一致性有所提高。我们识别出一个结构性原因：弱混沌成员的不变测度集中在低维集合上，而具有全支撑输出的模型无法表示这些测度，并且随着混沌增加，目标接近全维。将训练限制在强混沌成员上，使分类器在这些成员上的超出偶然水平的过剩量减少了约40%，并且这种受限条件模型达到或超过了在单一参数值上训练的模型。在这些成员上，生成的样本在重采样下限内再现了能量谱，在最大尺度上存在小的系统性偏差，并与保留数据的能量平衡相匹配。

英文摘要

The long-run statistics of chaotic dissipative partial differential equations are described by invariant measures, and when the equation carries a parameter these measures form a family whose members can differ in kind, from steady states to sustained spatiotemporal chaos. We ask whether a single generative model can cover such a family. We train a single conditional diffusion model once, on states of the Kuramoto-Sivashinsky equation at 160 values of its hyperviscosity, and evaluate it against held-out data at interleaved parameter values not seen in training, using a classifier two-sample test supported by spectral, geometric and tail statistics. Over most of the family, agreement improves as the dynamics become more chaotic. We identify a structural cause: the weakly chaotic members have invariant measures concentrated on low-dimensional sets, which a model with full support output cannot represent, and the target approaches full dimensionality as chaos increases. Restricting training to the strongly chaotic members reduces the classifier's excess over chance on those members by roughly 40%, and this restricted conditional model matches or exceeds models trained at single parameter values. On those members, generated samples reproduce energy spectra to within a resampling floor, with a small systematic deficit at the largest scales, and match the energy balance of held-out data.

发表机构

• University of Delaware（特拉华大学）

机构由 AI 辅助整理，请以论文原文为准。

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15 pages, 4 figures