A Deep Double Ritz Method for solving Partial Differential Equations

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. Such min-max problem is highly non-linear, and traditional methods often employ different mixed formulations to approximate it. Alternatively, it is possible to address the above saddle-point problem by employing Adversarial Neural Networks: one network approximates the global trial minimum, while another network seeks the test maximizer. However, this approach is numerically unstable due to a lack of continuity of the text maximizers with respect to the trial functions as we approach the exact 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. The resulting Deep Double Ritz Method combines two Neural Networks for approximating the trial and optimal test functions. Numerical results on several 1D diffusion and convection problems support the robustness of our method up to the approximability and trainability capacity of the networks and the optimizer.