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When using Neural Networks as trial functions to numerically solve PDEs, akey choice to be made is the loss function to be minimised, which shouldideally correspond to a norm of the error. In multiple problems, this errornorm coincides with--or is equivalent to--the $H^{-1}$-norm of the residual;however, it is often difficult to accurately compute it. This work assumesrectangular domains and proposes the use of a Discrete Sine/Cosine Transform toaccurately and efficiently compute the $H^{-1}$ norm. The resulting DeepFourier-based Residual (DFR) method efficiently and accurately approximatesolutions to PDEs. This is particularly useful when solutions lack $H^{2}$regularity and methods involving strong formulations of the PDE fail. Weobserve that the $H^1$-error is highly correlated with the discretised lossduring training, which permits accurate error estimation via the loss.

A Deep Fourier Residual Method for solving PDEs using Neural Networks