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

TL;DR

The paper introduces the Deep Double Ritz Method (D$^2$RM), a neural network-based approach for solving PDEs that reformulates residual minimization into a nested Ritz functional minimization to improve stability and robustness.

Contribution

It proposes a novel double Ritz minimization framework using neural networks to stabilize residual minimization in PDE solutions, addressing instability issues in min-max approaches.

Findings

Numerical experiments demonstrate robustness across diffusion and convection problems.

The method's success depends on network approximation capabilities and optimizer training.

D$^2$RM outperforms traditional residual minimization techniques in stability.

Abstract

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 different 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.