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

01/2019

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

Thursday 17th January 2019

Regazzoni, F.; Dedè, L.; Quarteroni, A.

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. One of the HF systems of ODEs here considered stems from the spatial discretization of a parabolic PDE, which sheds light on a promising field of application of the proposed technique.

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Submitted to Journal of Computational Physics