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

Highlights: • We propose a data-driven Model Order Reduction technique for time-dependent differential equations. • The reduced model is built solely on the basis of input-output pairs generated by the high-fidelity model. • We formulate the MOR problem as a maximum-likelihood problem in a class of candidate models. • Candidate models are represented by Artificial Neural Networks trained from the input-output pairs. • The technique is validated against large-scale ODEs, parabolic PDEs and hyperbolic PDEs. -- Abstract: 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.