Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations

We introduce physics informed neural networks - neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations. Depending on the nature and arrangement of the available data, we devise two distinct classes of algorithms, namely continuous time and discrete time models. The resulting neural networks form a new class of data-efficient universal function approximators that naturally encode any underlying physical laws as prior information. In this first part, we demonstrate how these networks can be used to infer solutions to partial differential equations, and obtain physics-informed surrogate models that are fully differentiable with respect to all input coordinates and free parameters. Introduction With the explosive growth of available data and computing resources, recent advances in machine learning and data analytics have yielded transfor-mative results across diverse scientific disciplines, including image recognition [1], natural language processing [2], cognitive science [3], and genomics [4]. In this small data regime, the vast majority of state-of-the art machine learning techniques (e.g., deep/- convolutional/recurrent neural networks) are lacking robustness and fail to provide any guarantees of convergence. At first sight, the task of training a deep learning algorithm to accurately identify a nonlinear map from a few - potentially very high-dimensional - input and output data pairs seems at best naive.