Data-Driven Finite Elements Methods: Machine Learning Acceleration of Goal-Oriented Computations

Abstract

We introduce the concept of data-driven finite element methods. These are finite-element discretizations of partial differential equations (PDEs) that resolve quantities of interest with striking accuracy, regardless of the underlying mesh size. The methods are obtained within a machine-learning framework during which the parameters defining the method are tuned against available training data. In particular, we use a stable parametric Petrov-Galerkin method that is equivalent to a minimal-residual formulation using a weighted norm. While the trial space is a standard finite element space, the test space has parameters that are tuned in an off-line stage. Finding the optimal test space therefore amounts to obtaining a goal-oriented discretization that is completely tailored towards the quantity of interest. As is natural in deep learning, we use an artificial neural network to define the parametric family of test spaces. Using numerical examples for the Laplacian and advection equation in one and two dimensions, we demonstrate that the data-driven finite element method has superior approximation of quantities of interest even on very coarse meshes

24 pages, 9 figures

References in corpus (2)

• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Data-driven discovery of PDEs in complex datasets

Cited by in corpus (7)

• On quadrature rules for solving Partial Differential Equations using Neural Networks
• A priori and a posteriori error estimates for the Deep Ritz method applied to the Laplace and Stokes problem
• A Deep Double Ritz Method (D${}^2$RM) for solving Partial Differential Equations using Neural Networks
• Goal-oriented adaptive finite element method for semilinear elliptic PDEs
• Neural Control of Discrete Weak Formulations: Galerkin, Least-Squares and Minimal-Residual Methods with Quasi-Optimal Weights
• A Posteriori Single- and Multi-Goal Error Control and Adaptivity for Partial Differential Equations
• Learning quantities of interest from parametric PDEs: An efficient neural-weighted Minimal Residual approach