利用对称性加速非线性降维建模，提升计算效率与稳定性。
Physics-informed reduced-order modelling with equivariant spectral submanifolds
- 将系统对称性融入谱子流形构造，实现更高效的降维。
- 在多个基准测试中显著降低计算耗时，且模型更鲁棒。
- 适合需要高效高精度非线性模型的工程与科学计算场景。
谱子流形（Spectral Submanifold, SSM）降维已成为一种数学严谨的非线性降维方法，能够捕捉线性方法（如动态模态分解，DMD）无法处理的动力学特征。然而，SSM 的计算仍存在高维系统下耗时过长的问题。本文提出等变谱子流形（equivariant SSM, eSSM）降维，首次将全阶模型的对称性显式引入降维过程。我们从理论上证明：SSM 本身是自然的等变子流形，其对应图和约化动力学继承了相应的诱导群作用。基于此框架，我们设计了一种新型等变 SSM 算法，利用对称性实现显著更快的计算速度，同时增强模型鲁棒性。我们在多个基准问题上验证了该方法的优势，包括来自科学共同任务框架（Common Task Framework for Science）的测试案例。
原文摘要 · Abstract (English)
Spectral submanifold (SSM) reduction has emerged as a mathematically principled route to reliable nonlinear reduced-order models, capturing dynamics beyond the reach of linear techniques such as Dynamic Mode Decomposition (DMD). The computation of SSMs, however, remains computationally expensive, particularly for high-dimensional systems. In this work, we introduce equivariant spectral submanifold (eSSM) reduction, a novel extension of the SSM framework that explicitly incorporates symmetries of the full-order model into the reduction process. We establish the mathematical foundations of this approach by showing that SSMs are naturally equivariant submanifolds and that the associated charts and reduced dynamics inherit the appropriate induced group actions. Building on this framework, we develop a novel equivariant SSM reduction algorithm that exploits these symmetries to achieve substantially faster computations while also improving model robustness. We demonstrate the advantages of this approach on several benchmark problems including a test from the Common Task Framework for Science.
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