Residual minimization is a widely used technique for solving Partial
Differential Equations in variational form. It minimizes the dual norm of the
residual, which naturally yields a saddle-point (min-max) problem over the
so-called trial and test spaces. In the context of neural networks, we can
address this min-max approach by employing one network to seek the trial
minimum, while another network seeks the test maximizers. However, the
resulting method is numerically unstable as we approach the trial solution. To
overcome this, we reformulate the residual minimization as an equivalent
minimization of a Ritz functional fed by optimal test functions computed from
another Ritz functional minimization. We call the resulting scheme the Deep
Double Ritz Method (D${}^{2}$ RM), which combines two neural networks for
approximating trial functions and optimal test functions along a nested double
Ritz minimization strategy. Numerical results on several 1D diffusion and
convection problems support the robustness of our method, up to the
approximation properties of the networks and the training capacity of the
optimizers.

PREPRINT

# A Deep Double Ritz Method (D${}^{2}$ RM) for solving Partial Differential
Equations using Neural Networks

Carlos Uriarte, David Pardo, Ignacio Muga, Judit Muñoz-Matute

Submitted on 7 November 2022, last revised on 17 November 2022

## Abstract

## Preprint

Comment: 26 pages

Subjects: Mathematics - Numerical Analysis; Computer Science - Machine Learning