Machine learning for fast and reliable solution of time dependent differential equations
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.
J. Comput. Phys.