A Deep Fourier Residual Method for solving PDEs using Neural Networks
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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 H^2 regularity and methods involving strong formulations of the PDE fail. We observe that the H^1-error is highly correlated with the discretised loss during training, which permits accurate error estimation via the loss.
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