Learning a PDE and its Solution + Underconstrained for AIChE

Tom S. Bertalan1, Felix Kemeth1, Tianqi Cui1 and Ioannis G. Kevrekidis1,2,3, (1)Chemical and Biomolecular Engineering, Johns Hopkins University, Baltimore, MD, (2)Program in Applied and Computational Mathematics, and Chemical and Biological Engineering, Johns Hopkins University, Princeton, NJ, (3)Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD

We describe and demonstrate an artificial neural network (ANN) approach for exploiting disjointed/patchwise data to learn simultaneously the law of a partial differential equation (PDE), and a solution which usefully interpolates between the patches and possibly also extrapolates beyond them. This work represents the synthesis of two previously separate efforts, one in which the PDE itself was learned [1], and one in which the interpolation, termed a "physically-informed neural network" (PINN), was learned for a known PDE [2]. The PINN framework can be thought of as the neural solution of a known PDE in space and time.

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We begin the exposition by describing our method for extracting a PDE from space-time data, with a sample application to learning the Kuramoto-Sivashiksky equation and the viscous Burgers equation. We then proceed to show how this network can be used in the PINN framework to produce long-range interpolations without recourse to a known governing PDE. In both parts, we additionally describe some relevant computational implementation details.

[1] Rico-Martínez, Krischer, Kevrekidis, Kube, and Hudson. “Discrete- vs Continuous-Time Nonlinear Signal Processing of Cu Electrodissolution Data.” Chem. Eng. Comm. (1992)

[2] Raissi, Perdikaris, Karniadakis. "Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations." (2017)