Multiscale and Nonlocal Learning for PDEs using Densely Connected RNNs
arXiv:2109.01790
Abstract
Learning time-dependent partial differential equations (PDEs) that govern evolutionary observations is one of the core challenges for data-driven inference in many fields. In this work, we propose to capture the essential dynamics of numerically challenging PDEs arising in multiscale modeling and simulation -- kinetic equations. These equations are usually nonlocal and contain scales/parameters that vary by several orders of magnitude. We introduce an efficient framework, Densely Connected Recurrent Neural Networks (DC-RNNs), by incorporating a multiscale ansatz and high-order implicit-explicit (IMEX) schemes into RNN structure design to identify analytic representations of multiscale and nonlocal PDEs from discrete-time observations generated from heterogeneous experiments. If present in the observed data, our DC-RNN can capture transport operators, nonlocal projection or collision operators, macroscopic diffusion limit, and other dynamics. We provide numerical results to demonstrate the advantage of our proposed framework and compare it with existing methods.
References in corpus (14)
- Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
- Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
- Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations
- PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network
- Deep Network Approximation for Smooth Functions
- Discovery of Nonlinear Multiscale Systems: Sampling Strategies and Embeddings
- Data-Driven Deep Learning of Partial Differential Equations in Modal Space
- Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems
- Machine Learning for Prediction with Missing Dynamics
- OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle
- Deep Multiscale Model Learning
- Reproducing Activation Function for Deep Learning
- Discovery of Dynamics Using Linear Multistep Methods
- The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimation