A Deep Double Ritz Method (D<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e3266" altimg="si244.svg"><mml:msup><mml:mrow /><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math>RM) for solving Partial Differential Equations using Neural Networks

Residual minimization is a widely used technique for solving Partial Differential Equations in variational form. It minimizes the dual norm of residual, which naturally yields saddle-point (min–max) problem over so-called trial and test spaces. In context neural networks, we can address this min–max approach by employing one network to seek minimum, while another seeks maximizers. However, resulting method numerically unstable as solution. To overcome this, reformulate residual an equivalent Ritz functional fed optimal functions computed from minimization. We call scheme Deep Double Method (D2RM), combines two networks approximating along nested double strategy. Numerical results on different diffusion convection problems support robustness our method, up approximation properties training capacity optimizers.