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