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This paper examines the coincidence of neural networks with numerical methodsfor solving spatiotemporal physical problems. Neural networks are used to learnpredictive numerical models from trajectory datasets from two well understood1D problems: the heat equation and the inviscid Burgers' equation. Coincidencewith established numerical methods is shown by demonstrating that a singlelayer convolutional neural network (CNN) converges to a traditional finitedifference stencil for the heat equation. However, a discriminator-basedadversarial training method, such as those used in generative adversarialnetworks (GANs), does not find the expected weights. A compact deep CNN isapplied to nonlinear Burgers' equation, where the models' architecture isreminiscent of existing winding finite volume methods. By searching overarchitectures and using multiple recurrent steps in the training loss, a modelis found that can integrate in time, recurring on its outputs, with similaraccuracy and stability to Godunov's method.

Studying Shallow and Deep Convolutional Neural Networks as Learned Numerical Schemes on the 1D Heat Equation and Burgers' Equation