Kaszás Bálint, Haller George
Institute for Mechanical Systems, ETH Zürich, Zurich, Switzerland.
Nat Commun. 2025 Jul 1;16(1):5722. doi: 10.1038/s41467-025-61252-9.
One of the very few mathematically rigorous nonlinear model reduction methods is the restriction of a dynamical system to a low-dimensional, sufficiently smooth, attracting invariant manifold. Such manifolds are usually found using local polynomial approximations and, hence, are limited by the unknown domains of convergence of their Taylor expansions. To address this limitation, we extend local expansions for invariant manifolds via Padé approximants, which re-express the Taylor expansions as rational functions for broader utility. This approach significantly expands the range of applicability of manifold-reduced models, enabling reduced modeling of global phenomena, such as large-scale oscillations and chaotic attractors of finite element models. We illustrate the power of globalized manifold-based model reduction on several equation-driven and data-driven examples from solid mechanics and fluid mechanics.
极少数数学上严格的非线性模型约简方法之一是将动力系统限制在低维、足够光滑的吸引不变流形上。此类流形通常通过局部多项式逼近找到，因此受到其泰勒展开式未知收敛域的限制。为了解决这一限制，我们通过帕德逼近扩展不变流形的局部展开式，将泰勒展开式重新表示为有理函数以获得更广泛的用途。这种方法显著扩展了流形约简模型的适用范围，能够对全局现象进行约简建模，例如有限元模型的大规模振荡和混沌吸引子。我们通过固体力学和流体力学中的几个方程驱动和数据驱动的例子来说明基于全局化流形的模型约简的强大功能。
