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We present a framework for recovering/approximating unknown time-dependentpartial differential equation (PDE) using its solution data. Instead ofidentifying the terms in the underlying PDE, we seek to approximate theevolution operator of the underlying PDE numerically. The evolution operator ofthe PDE, defined in infinite-dimensional space, maps the solution from acurrent time to a future time and completely characterizes the solutionevolution of the underlying unknown PDE. Our recovery strategy relies onapproximation of the evolution operator in a properly defined modal space,i.e., generalized Fourier space, in order to reduce the problem to finitedimensions. The finite dimensional approximation is then accomplished bytraining a deep neural network structure, which is based on residual network(ResNet), using the given data. Error analysis is provided to illustrate thepredictive accuracy of the proposed method. A set of examples of differenttypes of PDEs, including inviscid Burgers' equation that develops discontinuityin its solution, are presented to demonstrate the effectiveness of the proposedmethod.

Data-Driven Deep Learning of Partial Differential Equations in Modal Space