  This paper examines the coincidence of neural networks with numerical methods
for solving spatiotemporal physical problems. Neural networks are used to learn
predictive numerical models from trajectory datasets from two well understood
1D problems: the heat equation and the inviscid Burgers' equation. Coincidence
with established numerical methods is shown by demonstrating that a single
layer convolutional neural network (CNN) converges to a traditional finite
difference stencil for the heat equation. However, a discriminator-based
adversarial training method, such as those used in generative adversarial
networks (GANs), does not find the expected weights. A compact deep CNN is
applied to nonlinear Burgers' equation, where the models' architecture is
reminiscent of existing winding finite volume methods. By searching over
architectures and using multiple recurrent steps in the training loss, a model
is found that can integrate in time, recurring on its outputs, with similar
accuracy and stability to Godunov's method.
