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In recent years, numerical methods in industrial applications have evolved from a pure predictive tool towards a means for optimization and control. Since standard numerical analysis methods have become prohibitively costly in such…

Computational Physics · Physics 2021-04-23 Artūrs Bērziņš , Jan Helmig , Fabian Key , Stefanie Elgeti

We present a novel reduced order model (ROM) approach for parameterized time-dependent PDEs based on modern learning. The ROM is suitable for multi-query problems and is nonintrusive. It is divided into two distinct stages: A nonlinear…

Numerical Analysis · Mathematics 2020-11-24 Nikolaj T. Mücke , Sander M. Bohté , Cornelis W. Oosterlee

Natural convection in porous media is a highly nonlinear multiphysical problem relevant to many engineering applications (e.g., the process of $\mathrm{CO_2}$ sequestration). Here, we present a non-intrusive reduced order model of natural…

Computational Engineering, Finance, and Science · Computer Science 2023-08-30 T. Kadeethum , F. Ballarin , Y. Choi , D. O'Malley , H. Yoon , N. Bouklas

Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computational domains, etc.…

Numerical Analysis · Mathematics 2024-02-06 Zhanhong Ye , Xiang Huang , Hongsheng Liu , Bin Dong

Physics-based models often involve large systems of parametrized partial differential equations, where design parameters control various properties. However, high-fidelity simulations of such systems on large domains or with high grid…

Computational Physics · Physics 2025-05-15 Diba Behnoudfar

Nonintrusive projection-based reduced order models (ROMs) are essential for dynamics prediction in multi-query applications where access to the source of the underlying full order model (FOM) is unavailable; that is, FOM is a black-box.…

Computational Physics · Physics 2024-10-16 Aviral Prakash , Yongjie Jessica Zhang

In recent years, large-scale numerical simulations played an essential role in estimating the effects of explosion events in urban environments, for the purpose of ensuring the security and safety of cities. Such simulations are…

Constructing reduced-order models (ROMs) capable of efficiently predicting the evolution of high-dimensional, parametric systems is crucial in many applications in engineering and applied sciences. A popular class of projection-based ROMs…

Machine Learning · Computer Science 2026-03-17 Nicolò Botteghi , Silke Glas , Christoph Brune

In recent years, data-driven deep learning models have gained significant interest in the analysis of turbulent dynamical systems. Within the context of reduced-order models (ROMs), convolutional autoencoders (CAEs) pose a universally…

The popularity of deep convolutional autoencoders (CAEs) has engendered new and effective reduced-order models (ROMs) for the simulation of large-scale dynamical systems. Despite this, it is still unknown whether deep CAEs provide superior…

Machine Learning · Computer Science 2022-03-14 Anthony Gruber , Max Gunzburger , Lili Ju , Zhu Wang

Non-affine parametric dependencies, nonlinearities and advection-dominated regimes of the model of interest can result in a slow Kolmogorov n-width decay, which precludes the realization of efficient reduced-order models based on linear…

Numerical Analysis · Mathematics 2022-03-02 Francesco Romor , Giovanni Stabile , Gianluigi Rozza

Reduced Order Models (ROMs) are of considerable importance in many areas of engineering in which computational time presents difficulties. Established approaches employ projection-based reduction such as Proper Orthogonal Decomposition,…

Numerical Analysis · Mathematics 2023-04-26 Thomas Simpson , Konstantinos Vlachas , Anthony Garland , Nikolaos Dervilis , Eleni Chatzi

Kinetic equations are crucial for modeling non-equilibrium phenomena, but their computational complexity is a challenge. This paper presents a data-driven approach using reduced order models (ROM) to efficiently model non-equilibrium flows…

Fluid Dynamics · Physics 2023-10-09 Julian Koellermeier , Philipp Krah , Julius Reiss , Zachary Schellin

The coupling of Proper Orthogonal Decomposition (POD) and deep learning-based ROMs (DL-ROMs) has proved to be a successful strategy to construct non-intrusive, highly accurate, surrogates for the real time solution of parametric nonlinear…

Numerical Analysis · Mathematics 2024-05-15 Simone Brivio , Stefania Fresca , Andrea Manzoni

Reduced order modeling (ROM) provides an efficient framework to compute solutions of parametric problems. Basically, it exploits a set of precomputed high-fidelity solutions --- computed for properly chosen parameters, using a full-order…

Numerical Analysis · Mathematics 2019-11-19 Nicola Demo , Marco Tezzele , Gianluigi Rozza

Reduced-order modeling (ROM) commonly refers to the construction, based on a few solutions (referred to as snapshots) of an expensive discretized partial differential equation (PDE), and the subsequent application of low-dimensional…

Numerical Analysis · Mathematics 2019-05-22 Martin Hess , Alessandro Alla , Annalisa Quaini , Gianluigi Rozza , Max Gunzburger

This study presents a collection of purely data-driven workflows for constructing reduced-order models (ROMs) for distributed dynamical systems. The ROMs we focus on, are data-assisted models inspired by, and templated upon, the theory of…

The use of deep learning has become increasingly popular in reduced-order models (ROMs) to obtain low-dimensional representations of full-order models. Convolutional autoencoders (CAEs) are often used to this end as they are adept at…

This paper presents a novel non-linear model reduction method: Probabilistic Manifold Decomposition (PMD), which provides a powerful framework for constructing non-intrusive reduced-order models (ROMs) by embedding a high-dimensional system…

Numerical Analysis · Mathematics 2026-01-09 Jiaming Guo , Dunhui Xiao

Simulations of large-scale dynamical systems require expensive computations. Low-dimensional parametrization of high-dimensional states such as Proper Orthogonal Decomposition (POD) can be a solution to lessen the burdens by providing a…

Dynamical Systems · Mathematics 2024-10-30 Yongho Kim , Jan Heiland