Data Driven Governing Equations Approximation Using Deep Neural Networks

By Tong Qin, Kailiang Wu, Dongbin Xiu. Journal of Computational Physics, 2018

308 citations recorded by the source corpus.

Abstract We present a numerical framework for approximating unknown governing equations using observation data and deep neural networks (DNN). In particular, we propose to use residual network (ResNet) as the basic building block for equation approximation. We demonstrate that the ResNet block can be considered as a one-step method that is exact in temporal integration. We then present two multi-step methods, recurrent ResNet (RT-ResNet) method and recursive ReNet (RS-ResNet) method. The RT-ResNet is a multi-step method on uniform time steps, whereas the RS-ResNet is an adaptive multi-step method using variable time steps. All three methods presented here are based on integral form of the underlying dynamical system. As a result, they do not require time derivative data for equation recovery and can cope with relatively coarsely distributed trajectory data. Several numerical examples are presented to demonstrate the performance of the methods.

Subfields: Artificial Intelligence, Control and Systems Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/74f243e8cc5cfb7f15387a51c27a24c97613c8ed

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Control and Systems Engineering

In Statistical and Nonlinear Physics

Citations in the corpus

Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators (2019, methodology): “The network architectures they employed includes FNNs [24], recurrent neural networks (RNNs) [20], reservoir computing [20], residual networks [23], autoencoder [9], neural ordinary diﬀerential equations [6], and neural jump stochastic diﬀerential equations [12].”
Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks (2019, methodology): “Furthermore, deep ResNets have, recently, been sucessfully applied to the task of data driven approximation of governing equations of dynamical systems in [46].”
Stiff-PINN: Physics-Informed Neural Network for Stiff Chemical Kinetics (2020, methodology): “17 Reference [1] T.”
Efficient training of physics‐informed neural networks via importance sampling (2021, unknown): “With that in mind, this paper takes a step forward toward improving the computational eﬃciency of PINNs for solving PDEs.”
WEAK SINDY FOR PARTIAL DIFFERENTIAL EQUATIONS (2020, background): “More recently, the authors of [15] and [26] combine neural network-based recovery schemes with integral and abstract evolution equations to recover PDE dynamics.”
PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network (2018, background): “An interesting fact is that some popular networks in computer vision, such as ResNet[15, 16], have close relationship with ODEs/PDEs and can be naturally merged with traditional computational mathematics in various tasks [17, 18, 19, 20, 21, 22, 23, 24, 25, 26].”
A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate (2021, methodology): “applied the deep residual network and observation data to approximate unknown governing differential equations [44].”
Disentangling Physical Dynamics From Unknown Factors for Unsupervised Video Prediction (2020, background): “Euler, Runge-Kutta) and residual neural networks [71, 37, 8, 78], several specific architectures were designed for predicting and identifying dynamical systems [15, 35, 47].”