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This work formulates a new approach to reduced modeling of parameterized, time-dependent partial differential equations (PDEs). The method employs Operator Inference, a scientific machine learning framework combining data-driven learning…
In this paper, we propose a network model, the multiclass classification-based reduced order model (MC-ROM), for solving time-dependent parametric partial differential equations (PPDEs). This work is inspired by the observation of applying…
Reduced basis approximations of Optimal Control Problems (OCPs) governed by steady partial differential equations (PDEs) with random parametric inputs are analyzed and constructed. Such approximations are based on a Reduced Order Model,…
Solving optimal control problems for transport-dominated partial differential equations (PDEs) can become computationally expensive, especially when dealing with high-dimensional systems. To overcome this challenge, we focus on developing…
We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…
In this paper, we propose novel proper orthogonal decomposition (POD)--based model reduction methods that effectively address the issue of inverse crime in solving parabolic inverse problems. Both the inverse initial value problems and…
Scientific and engineering problems often involve parametric partial differential equations (PDEs), such as uncertainty quantification, optimizations, and inverse problems. However, solving these PDEs repeatedly can be prohibitively…
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.…
Projection-based reduced order models are effective at approximating parameter-dependent differential equations that are parametrically separable. When parametric separability is not satisfied, which occurs in both linear and nonlinear…
Solving optimal control problems for transport-dominated partial differential equations (PDEs) can become computationally expensive, especially when dealing with high-dimensional systems. To overcome this challenge, we focus on developing…
Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often…
This paper introduces a reduced-order modeling approach based on finite volume methods for hyperbolic systems, combining Proper Orthogonal Decomposition (POD) with the Discrete Empirical Interpolation Method (DEIM) and Proper Interval…
This work studies a variational formulation and numerical solution of a regularized morphoelasticity problem of shape evolution. The foundation of our analysis is based on the governing equations of linear elasticity, extended to account…
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…
In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems by time domain decomposition. The available algorithms enabling such decompositions present severe efficiency limitations and are an…
Partial differential equations (PDE) often involve parameters, such as viscosity or density. An analysis of the PDE may involve considering a large range of parameter values, as occurs in uncertainty quantification, control and…
We consider the problem of estimating parameters in large-scale weakly nonlinear inverse problems for which the underlying governing equations is a linear, time-dependent, parabolic partial differential equation. A major challenge in…
The present work proposes a well-balanced finite volume-type numerical method for the solution of non-conservative hyperbolic partial differential equations (PDEs) with source terms. The method is characterized, first, by the use of a…
We propose a numerical method to solve parameter-dependent hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of…
Hyperbolic partial differential equations (PDEs) cover a wide range of interesting phenomena, from human and hearth-sciences up to astrophysics: this unavoidably requires the treatment of many space and time scales in order to describe at…