Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations

A long-standing problem at the interface of artificial intelligence and mathematics is to devise an algorithm capable of achieving human level even superhuman proficiency in transforming observed data into predictive models of the physical world. In the current era of abundance of and advanced machine learning capabilities, the natural question arises: can we automatically uncover the underlying laws of physics from-dimensional data generated from experiments? In this work, we put forth a learning approach for discovering nonlinear partial differential equations scattered and potentially noisy observations in space and time. , we approximate the unknown solution as well as the nonlinear by two deep neural networks. The first network acts as a prior on the solution and essentially enables us to avoid numerical differentiations are inherently ill-conditioned and unstable. The second network the nonlinear dynamics and helps us distill the mechanisms that the evolution of a given spatiotemporal data-set. We test the of our approach for several benchmark problems spanning a number scientific domains and demonstrate how the proposed framework can help us learn the underlying dynamics and forecast future states of the. In particular, we study the Burgers', Korteweg-de Vries (KdV), -Sivashinsky, nonlinear Schr\\"odinger, and Navier-Stokes equations.