Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Space

Abstract

We present a numerical framework for deep neural network (DNN) modeling of unknown time-dependent partial differential equations (PDE) using their trajectory data. Unlike the recent work of [Wu and Xiu, J. Comput. Phys. 2020], where the learning takes place in modal/Fourier space, the current method conducts the learning and modeling in physical space and uses measurement data as nodal values. We present a DNN structure that has a direct correspondence to the evolution operator of the underlying PDE, thus establishing the existence of the DNN model. The DNN model also does not require any geometric information of the data nodes. Consequently, a trained DNN defines a predictive model for the underlying unknown PDE over structureless grids. A set of examples, including linear and nonlinear scalar PDE, system of PDEs, in both one dimension and two dimensions, over structured and unstructured grids, are presented to demonstrate the effectiveness of the proposed DNN modeling. Extension to other equations such as differential-integral equations is also discussed.

References in corpus (11)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
• DeepXDE: A deep learning library for solving differential equations
• Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations
• Data Driven Governing Equations Approximation Using Deep Neural Networks
• Deep learning of dynamics and signal-noise decomposition with time-stepping constraints
• Data-Driven Deep Learning of Partial Differential Equations in Modal Space
• Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems
• Numerical Aspects for Approximating Governing Equations Using Data
• Structure-preserving Method for Reconstructing Unknown Hamiltonian Systems from Trajectory Data
• IDENT: Identifying Differential Equations with Numerical Time evolution