Neural Gray-Box Identification of Nonlinear Partial Differential Equations

Abstract

Many branches of the modern computational science and engineering are based on numerical simulations, for which we must prepare appropriate equations that well reflect the behavior of real-world phenomena and numerically solve them. For these purposes, we may utilize the data-driven identification and simulation technique of nonlinear partial differential equations (NPDEs) using deep neural networks (DNNs). A potential issue of the DNN-based identification and simulation in practice is the high variance due to the complexity of DNNs. To alleviate it, we propose a simple yet efficient way to incorporate prior knowledge of phenomena. Specifically, we can often anticipate what kinds of terms are present in a part of an appropriate NPDE, which should be utilized as prior knowledge for identifying the remaining part of the NPDE. To this end, we design DNN’s inputs and the loss function for identification according to the prior knowledge. We present the results of the experiments conducted using three different types of NPDEs: the Korteweg–de Vries equation, the Navier–Stokes equation, and the Kuramoto–Sivashinsky equation. The experimental results show the effectiveness of the proposed method, i.e., utilizing known terms of an NPDE.