Machine learning for fast and reliable solution of time-dependent differential equations

JOURNAL OF COMPUTATIONAL PHYSICS

卷:397

Article

Regazzoni, F.¹  Dede, L.¹  Quarteroni, A.^1,2 

[1] Politecn Milan, MOX Dipartimento Matemat, Pzza Leonardo da Vinci 32, I-20133 Milan, Italy

[2] Ecole Polytech Fed Lausanne, Math Inst, Av Piccard, CH-1015 Lausanne, Switzerland

关键词: Machine learning;    Differential equations;    Model order reduction;    System identification;    Artificial neural networks;    Data-driven modeling;   

DOI  :  10.1016/j.jcp.2019.07.050

来源: Elsevier

【 摘 要 】

We propose a data-driven Model Order Reduction (MOR) technique, based on Artificial Neural Networks (ANNs), applicable to dynamical systems arising from Ordinary Differential Equations (ODEs) or time-dependent Partial Differential Equations (PDEs). Unlike model-based approaches, the proposed approach is non-intrusive since it just requires a collection of input-output pairs generated through the high-fidelity (HF) ODE or PDE model. We formulate our model reduction problem as a maximum-likelihood problem, in which we look for the model that minimizes, in a class of candidate models, the error on the available input-output pairs. Specifically, we represent candidate models by means of ANNs, which we train to learn the dynamics of the HF model from the training input-output data. We prove that ANN models are able to approximate every time-dependent model described by ODEs with any desired level of accuracy. We test the proposed technique on different problems, including the model reduction of two large-scale models. Two of the HF systems of ODEs here considered stem from the spatial discretization of a parabolic and an hyperbolic PDE respectively, which sheds light on a promising field of application of the proposed technique. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free