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.