A Deep Fourier Residual method for solving PDEs using Neural Networks

When using Neural Networks as trial functions to numerically solve PDEs, a key choice be made is the loss function minimised, which should ideally correspond norm of error. In multiple problems, this error coincides with – or equivalent H−1-norm residual; however, it often difficult accurately compute it. This work assumes rectangular domains and proposes use Discrete Sine/Cosine Transform efficiently H−1 norm. The resulting Deep Fourier-based Residual (DFR) method approximate solutions PDEs. particularly useful when lack H2 regularity methods involving strong formulations PDE fail. We observe that H1-error highly correlated discretised during training, permits accurate estimation via loss.