Carlos Uriarte

Basque Center for Applied Mathematics (BCAM), and Universidad del Paı́s Vasco/Euskal Herriko Unibertsitatea (UPV/EHU), Spain 

Residual minimization is a widely used technique for solving Partial Differential Equations in

variational form. It minimizes the residual over the trial functions on the dual norm of the

test space, which naturally yields a saddle-point (min-max) problem. Such min-max problem

is highly non-linear, and traditional methods often employ different discrete mixed formulations

to approximate it. Alternatively, it is possible to address the above saddle-point problem by

employing Neural Networks. Specifically, (Generative) Adversarial Networks can handle this task

by employing one network to find the global trial minimum, and another network to mimic the

test maximizer [1,2]. However, a straightforward implementation of this approach turns out to

be numerically unrobust due to the high non-uniqueness of the test maximizer for the global trial

minimum. In this work, we reformulate the residual minimization as an equivalent minimization

of a quadratic Ritz functional fed by optimal test functions [3], which can be computed from

another Ritz functional minimization. We implement this using two Neural Networks combined

into a nested optimization of trial and test problems. We test the resulting Deep Double Ritz

Method on several 1D diffusion and convection problems.

[1] Yang, Y., Bao, G., Ye, X., and Zhou, H., Weak adversarial networks for high-dimensional partial differential equations, J. Comput. Phys. 411 (2020), article no. 109409.

[2] Bao, G., Ye, X., Zang, Y., and Zhou, H., Numerical solution of inverse problems by weak adversarial networks, Inverse Problems 36 (2020), article no. 115003.

[3] Demkowicz, L. and Gopalakrishnan, J., A class of discontinuous Petrov-Galerkin methods. Part II. Optimal test functions, Numer. Methods for Partial Differ. Equ. 21 (2011), pp. 70–105.