Related papers: A data-driven approach for multiscale elliptic PDE…
We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks. Motivated by the universal approximation property of this type of networks, both methods extend the…
Given input-output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically-rigorous scheme for learning the associated Green's function $G$. By exploiting the hierarchical low-rank…
In recent years, there has been a growing interest in leveraging deep learning and neural networks to address scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods…
This work derives a residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control…
This paper is interested in developing reduced order models (ROMs) for repeated simulation of fractional elliptic partial differential equations (PDEs) for multiple values of the parameters (e.g., diffusion coefficients or fractional…
The task of simultaneously reconstructing multiple physical coefficients in partial differential equations (PDEs) from observed data is ubiquitous in applications. In this work, we propose an integrated data-driven and model-based iterative…
Discovering hidden partial differential equations (PDEs) and operators from data is an important topic at the frontier between machine learning and numerical analysis. This doctoral thesis introduces theoretical results and deep learning…
In this paper, we propose an offline-online strategy based on the Localized Orthogonal Decomposition (LOD) method for elliptic multiscale problems with randomly perturbed diffusion coefficient. We consider a periodic deterministic…
Dynamic Mode Decomposition (DMD) is a data-driven technique to identify a low dimensional linear time invariant dynamics underlying high-dimensional data. For systems in which such underlying low-dimensional dynamics is time-varying, a…
The recent years have seen a beautiful breakthrough culminating in a comprehensive understanding of certain scale-invariant properties of $n-1$ dimensional sets across analysis, geometric measure theory, and PDEs. The present paper surveys…
We propose a generalized multiscale finite element method (GMsFEM) based on clustering algorithm to study the elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the…
In this paper, we propose a data-driven framework for constructing efficient approximate inverse preconditioners for elliptic partial differential equations (PDEs) by learning the Green's function of the underlying operator with neural…
We consider the computation of averaged coefficients for the homogenization of elliptic partial differential equations. In this problem, like in many multiscale problems, a large number of similar computations parametrized by the…
In this paper, we consider an elliptic eigenvalue problem with multiscale, randomly perturbed coefficients. For an efficient and accurate approximation of the solutions for many different realizations of the coefficient, we propose a…
There is an intimate connection between numerical upscaling of multiscale PDEs and scattered data approximation of heterogeneous functions: the coarse variables selected for deriving an upscaled equation (in the former) correspond to the…
Recent developments in mechanical, aerospace, and structural engineering have driven a growing need for efficient ways to model and analyse structures at much larger and more complex scales than before. While established numerical methods…
The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length and timescales. Often, it is computationally intractable to resolve the finest features…
In this work, we present a POD-greedy reduced basis method for parabolic partial differential equations (PDEs), based on the least squares space-time formulation proposed in [Hinze, Kahle, Stahl, A least-squares space-time approach for…
In this work, we present an adjoint-based method for discovering the underlying governing partial differential equations (PDEs) given data. The idea is to consider a parameterized PDE in a general form and formulate a PDE-constrained…
We address the challenge of dimension reduction in the discrete-time optimal control problem which is solved repeatedly online within the framework of model predictive control. Our study demonstrates that a reduced-order approach, aimed at…