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Many applications in computational physics involve approximating problems with microstructure, characterized by multiple spatial scales in their data. However, these numerical solutions are often computationally expensive due to the need to…

Numerical Analysis · Mathematics 2023-09-15 Piermario Vitullo , Alessio Colombo , Nicola Rares Franco , Andrea Manzoni , Paolo Zunino

We propose a non-intrusive Deep Learning-based Reduced Order Model (DL-ROM) capable of capturing the complex dynamics of mechanical systems showing inertia and geometric nonlinearities. In the first phase, a limited number of high fidelity…

Numerical Analysis · Mathematics 2021-11-25 Stefania Fresca , Giorgio Gobat , Patrick Fedeli , Attilio Frangi , Andrea Manzoni

This paper aims at reviewing nonlinear methods for model order reduction of structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear based…

Numerical Analysis · Mathematics 2022-05-26 Cyril Touzé , Alessandra Vizzaccaro , Olivier Thomas

Structural dynamics models with nonlinear stiffness appear, for example, when analyzing systems with nonlinear material behavior or undergoing large deformations. For complex systems, these models become too large for real-time applications…

Numerical Analysis · Mathematics 2026-01-14 Pascal den Boef , Diana Manvelyan , Joseph Maubach , Wil Schilders , Nathan van de Wouw

Nonlinear manifold learning (ML) based reduced-order models (ROMs) can substantially improve the quality of nonlinear flow-field modeling. However, noise and the lack of physical information often distort the dimensionality-reduction…

Fluid Dynamics · Physics 2026-01-21 Weiji Wang , Chunlin Gong , Xuyi Jia , Chunna Li

We present an efficient data-driven regression approach for constructing reduced-order models (ROMs) of reaction-diffusion systems exhibiting pattern formation. The ROMs are learned non-intrusively from available training data of physically…

Pattern Formation and Solitons · Physics 2025-08-12 Alessandro Alla , Rudy Geelen , Hannah Lu

The simulation of electric rotating machines is both computationally expensive and memory intensive. To overcome these costs, model order reduction techniques can be applied. The focus of this contribution is especially on machines that…

Numerical Analysis · Mathematics 2017-05-11 Zeger Bontinck , Oliver Lass , Sebastian Schöps , Oliver Rain

State estimation is key to both analyzing physical mechanisms and enabling real-time control of fluid flows. A common estimation approach is to relate sensor measurements to a reduced state governed by a reduced-order model (ROM). (When…

Fluid Dynamics · Physics 2020-06-10 Nirmal J. Nair , Andres Goza

In this paper, five different approaches for reduced-order modeling of brittle fracture in geomaterials, specifically concrete, are presented and compared. Four of the five methods rely on machine learning (ML) algorithms to approximate…

Computational Engineering, Finance, and Science · Computer Science 2018-06-07 A. Hunter , B. A. Moore , M. K. Mudunuru , V. T. Chau , R. L. Miller , R. B. Tchoua , C. Nyshadham , S. Karra , D. O. Malley , E. Rougier , H. S. Viswanathan , G. Srinivasan

In this paper, we propose an equation-based parametric Reduced Order Model (ROM), whose accuracy is improved with data-driven terms added into the reduced equations. These additions have the aim of reintroducing contributions that in…

Numerical Analysis · Mathematics 2024-06-07 Anna Ivagnes , Giovanni Stabile , Gianluigi Rozza

Projection-based model reduction has become a popular approach to reduce the cost associated with integrating large-scale dynamical systems so they can be used in many-query settings such as optimization and uncertainty quantification. For…

Numerical Analysis · Mathematics 2020-08-26 Han Gao , Jian-Xun Wang , Matthew J. Zahr

Reduced order modeling (ROM) techniques are numerical methods that approximate the solution of parametric partial differential equation (PDE) by properly combining the high-fidelity solutions of the problem obtained for several…

Numerical Analysis · Mathematics 2023-08-08 M. Girfoglio , L. Scandurra , F. Ballarin , G. Infantino , F. Nicolò , A. Montalto , G. Rozza , R. Scrofani , M. Comisso , F. Musumeci

In this effort we propose a data-driven learning framework for reduced order modeling of fluid dynamics. Designing accurate and efficient reduced order models for nonlinear fluid dynamic problems is challenging for many practical…

Computational Physics · Physics 2018-12-05 Xuping Xie , Guannan Zhang , Clayton G. Webster

Model order reduction (MOR) involves offering low-dimensional models that effectively approximate the behavior of complex high-order systems. Due to potential model complexities and computational costs, designing controllers for…

Systems and Control · Electrical Eng. & Systems 2025-02-04 Behrad Samari , Amy Nejati , Abolfazl Lavaei

Traditional projection-based reduced-order modeling approximates the full-order model by projecting it onto a linear subspace. With a fast-decaying Kolmogorov $n$-width of the solution manifold, the resulting reduced-order model (ROM) can…

Numerical Analysis · Mathematics 2026-03-27 Lijie Ji , Sabrina Rashid , Yanlai Chen , Zhu Wang

This work presents a non-intrusive model reduction method to learn low-dimensional models of dynamical systems with non-polynomial nonlinear terms that are spatially local and that are given in analytic form. In contrast to state-of-the-art…

Numerical Analysis · Mathematics 2020-10-28 Peter Benner , Pawan Goyal , Boris Kramer , Benjamin Peherstorfer , Karen Willcox

Reduced Order Modelling (ROM) has been widely used to create lower order, computationally inexpensive representations of higher-order dynamical systems. Using these representations, ROMs can efficiently model flow fields while using…

Fluid Dynamics · Physics 2021-10-13 Pranshu Pant , Ruchit Doshi , Pranav Bahl , Amir Barati Farimani

Mechanical systems are often characterized only by their response to certain loads known from experiments or simulations. The obtained data can be used for various purposes: system analysis, design of mathematical models, or construction of…

Dynamical Systems · Mathematics 2026-01-05 Yevgeniya Filanova , Igor Pontes Duff , Pawan Goyal , Peter Benner

Reduced Order Models (ROMs) form essential tools across engineering domains by virtue of their function as surrogates for computationally intensive digital twinning simulators. Although purely data-driven methods are available for ROM…

Computational Engineering, Finance, and Science · Computer Science 2025-04-14 Konstantinos Vlachas , Thomas Simpson , Anthony Garland , D. Dane Quinn , Charbel Farhat , Eleni Chatzi

We present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The…

Numerical Analysis · Mathematics 2026-05-27 Rudy Geelen , Laura Balzano , Stephen Wright , Karen Willcox