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

A long-standing problem at the interface of artificial intelligence and\napplied mathematics is to devise an algorithm capable of achieving human level\nor even superhuman proficiency in transforming observed data into predictive\nmathematical models of the physical world. In the current era of abundance of\ndata and advanced machine learning capabilities, the natural question arises:\nHow can we automatically uncover the underlying laws of physics from\nhigh-dimensional data generated from experiments? In this work, we put forth a\ndeep learning approach for discovering nonlinear partial differential equations\nfrom scattered and potentially noisy observations in space and time.\nSpecifically, we approximate the unknown solution as well as the nonlinear\ndynamics by two deep neural networks. The first network acts as a prior on the\nunknown solution and essentially enables us to avoid numerical differentiations\nwhich are inherently ill-conditioned and unstable. The second network\nrepresents the nonlinear dynamics and helps us distill the mechanisms that\ngovern the evolution of a given spatiotemporal data-set. We test the\neffectiveness of our approach for several benchmark problems spanning a number\nof scientific domains and demonstrate how the proposed framework can help us\naccurately learn the underlying dynamics and forecast future states of the\nsystem. In particular, we study the Burgers', Korteweg-de Vries (KdV),\nKuramoto-Sivashinsky, nonlinear Schr\\"{o}dinger, and Navier-Stokes equations.\n