量化局部时间降阶建模(ql-ROM)

IF 7.3 1区 工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY
Antonio Colanera , Luca Magri
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引用次数: 0

摘要

时空混沌系统,如一些非线性偏微分方程的解,是向低维流形演化的动力系统。该流形具有复杂的几何形状和非均匀密度,这使得单个(全局)非线性降阶模型(ROM)的设计具有挑战性。在本文中,我们将扭转这一局面。我们不是用单一模型对流形建模,而是将流形划分为簇,在簇中对动态进行局部建模。这就产生了量化的局部降阶模型(ql-ROM),它包括:(1)通过无监督聚类对流形进行量化;(ii)为每个集群构建侵入式只读存储器;(iii)将局部模型的切换与基和赋值函数的变化联系起来。我们在两个非线性偏微分方程上测试了该方法,即Kuramoto-Sivashinsky和2D Navier-Stokes方程(Kolmogorov流),跨越爆发,混沌,准周期和湍流状态。局部模型通过伽辽金投影在以聚类质心为中心的局部主方向上建立。基于簇的接近度,通过切换局部rom来建立动态模型。与全局ROMs (g-ROMs)相比,本文提出的ql-ROM框架具有三个优势:(i)数值稳定性;(ii)提高短期预测的时间精度;(iii)准确预测长期统计数据,如能谱和概率分布。相对于g- rom,计算开销是最小的。所提出的框架保留了侵入式基于投影的rom的可解释性和简单性,同时克服了它们在建模复杂、高维、非线性动力学方面的局限性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Quantized local reduced-order modeling in time (ql-ROM)
Spatiotemporally chaotic systems, such as the solutions of some nonlinear partial differential equations, are dynamical systems that evolve toward a lower dimensional manifold. This manifold has an intricate geometry with heterogeneous density, which makes the design of a single (global) nonlinear reduced-order model (ROM) challenging. In this paper, we turn this around. Instead of modeling the manifold with one single model, we partition the manifold into clusters within which the dynamics are locally modeled. This results in a quantized local reduced-order model (ql-ROM), which consists of (i) quantizing the manifold via unsupervised clustering; (ii) constructing intrusive ROMs for each cluster; and (iii) connecting the switching of local models with a change of basis and assignment functions. We test the method on two nonlinear partial differential equations, i.e., the Kuramoto-Sivashinsky and 2D Navier-Stokes equations (Kolmogorov flow), across bursting, chaotic, quasiperiodic, and turbulent regimes. The local models are built via Galerkin projection onto the local principal directions, which are centered on the cluster centroids. The dynamics are modeled by switching the local ROMs based on the cluster proximity. The proposed ql-ROM framework has three advantages over global ROMs (g-ROMs): (i) numerical stability, (ii) improved short-term prediction accuracy in time, and (iii) accurate prediction of long-term statistics, such as energy spectra and probability distributions. The computational overhead is minimal with respect to g-ROMs. The proposed framework retains the interpretability and simplicity of intrusive projection-based ROMs, whilst overcoming their limitations in modeling complex, high-dimensional, nonlinear dynamics.
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来源期刊
CiteScore
12.70
自引率
15.30%
发文量
719
审稿时长
44 days
期刊介绍: Computer Methods in Applied Mechanics and Engineering stands as a cornerstone in the realm of computational science and engineering. With a history spanning over five decades, the journal has been a key platform for disseminating papers on advanced mathematical modeling and numerical solutions. Interdisciplinary in nature, these contributions encompass mechanics, mathematics, computer science, and various scientific disciplines. The journal welcomes a broad range of computational methods addressing the simulation, analysis, and design of complex physical problems, making it a vital resource for researchers in the field.
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