Data-Driven Learning of Nonautonomous Systems

Qin, Tong; Chen, Zhen; Jakeman, John D.; Xiu, Dongbin

doi:10.1137/20m1342859

Data-Driven Learning of Nonautonomous Systems

Journal Article · Thu May 06 00:00:00 EDT 2021 · SIAM Journal on Scientific Computing

DOI:https://doi.org/10.1137/20m1342859· OSTI ID:1883180

^[1]; Chen, Zhen ^[2]; Jakeman, John D. ^[3]; Xiu, Dongbin ^[2]

Univ. of Michigan, Ann Arbor, MI (United States)
The Ohio State Univ., Columbus, OH (United States)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

In this work, we present a numerical framework for recovering unknown nonautonomous dynamical systems with time-dependent inputs. To circumvent the difficulty presented by the nonautonomous nature of the system, our method transforms the solution state into piecewise integration of the system over a discrete set of time instances. The time-dependent inputs are then locally parameterized by using a proper model, for example, polynomial regression, in the pieces determined by the time instances. This transforms the original system into a piecewise parametric system that is locally time invariant. We then design a deep neural network structure to learn the local models. Once the network model is constructed, it can be iteratively used over time to conduct global system prediction. We provide theoretical analysis of our algorithm and present a number of numerical examples to demonstrate the effectiveness of the method.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE National Nuclear Security Administration (NNSA); US Air Force Office of Scientific Research (AFOSR)

Grant/Contract Number:: NA0003525

OSTI ID:: 1883180

Report Number(s):: SAND2021-11696J; 699538

Journal Information:: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 3 Vol. 43; ISSN 1064-8275

Publisher:: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement

Country of Publication:: United States

Language:: English

References (19)

Spectral Properties of Dynamical Systems, Model Reduction and Decompositions Mezić, Igor Nonlinear Dynamics, Vol. 41, Issue 1-3 https://doi.org/10.1007/s11071-005-2824-x	journal	August 2005
Data-driven operator inference for nonintrusive projection-based model reduction Peherstorfer, Benjamin; Willcox, Karen Computer Methods in Applied Mechanics and Engineering, Vol. 306 https://doi.org/10.1016/j.cma.2016.03.025	journal	July 2016
Sparse Identification of Nonlinear Dynamics with Control (SINDYc)**SLB acknowledges support from the U.S. Air Force Center of Excellence on Nature Inspired Flight Technologies and Ideas (FA9550-14-1-0398). JLP thanks Bill and Melinda Gates for their active support of the Institute of Disease Modeling and their sponsorship through the Global Good Fund. JNK acknowledges support from the U.S. Air Force Office of Scientific Research (FA9550-09-0174). Brunton, Steven L.; Proctor, Joshua L.; Kutz, J. Nathan IFAC-PapersOnLine, Vol. 49, Issue 18 https://doi.org/10.1016/j.ifacol.2016.10.249	journal	January 2016
Machine learning of linear differential equations using Gaussian processes Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em Journal of Computational Physics, Vol. 348 https://doi.org/10.1016/j.jcp.2017.07.050	journal	November 2017
Numerical aspects for approximating governing equations using data Wu, Kailiang; Xiu, Dongbin Journal of Computational Physics, Vol. 384 https://doi.org/10.1016/j.jcp.2019.01.030	journal	May 2019
Data driven governing equations approximation using deep neural networks Qin, Tong; Wu, Kailiang; Xiu, Dongbin Journal of Computational Physics, Vol. 395 https://doi.org/10.1016/j.jcp.2019.06.042	journal	October 2019
Deep learning of dynamics and signal-noise decomposition with time-stepping constraints Rudy, Samuel H.; Nathan Kutz, J.; Brunton, Steven L. Journal of Computational Physics, Vol. 396 https://doi.org/10.1016/j.jcp.2019.06.056	journal	November 2019
Data-driven deep learning of partial differential equations in modal space Wu, Kailiang; Xiu, Dongbin Journal of Computational Physics, Vol. 408 https://doi.org/10.1016/j.jcp.2020.109307	journal	May 2020
Dynamic mode decomposition of numerical and experimental data Schmid, Peter J. Journal of Fluid Mechanics, Vol. 656 https://doi.org/10.1017/S0022112010001217	journal	July 2010
Approximation theory of the MLP model in neural networks Pinkus, Allan Acta Numerica, Vol. 8 https://doi.org/10.1017/S0962492900002919	journal	January 1999
Discovering governing equations from data by sparse identification of nonlinear dynamical systems Brunton, Steven L.; Proctor, Joshua L.; Kutz, J. Nathan Proceedings of the National Academy of Sciences, Vol. 113, Issue 15 https://doi.org/10.1073/pnas.1517384113	journal	March 2016
"Coarse" stability and bifurcation analysis using time-steppers: A reaction-diffusion example Theodoropoulos, C.; Qian, Y. -H.; Kevrekidis, I. G. Proceedings of the National Academy of Sciences, Vol. 97, Issue 18 https://doi.org/10.1073/pnas.97.18.9840	journal	August 2000
Dynamic Mode Decomposition with Control Proctor, Joshua L.; Brunton, Steven L.; Kutz, J. Nathan SIAM Journal on Applied Dynamical Systems, Vol. 15, Issue 1 https://doi.org/10.1137/15M1013857	journal	January 2016
Generalizing Koopman Theory to Allow for Inputs and Control Proctor, Joshua L.; Brunton, Steven L.; Kutz, J. Nathan SIAM Journal on Applied Dynamical Systems, Vol. 17, Issue 1 https://doi.org/10.1137/16M1062296	journal	January 2018
Structure-Preserving Method for Reconstructing Unknown Hamiltonian Systems From Trajectory Data Wu, Kailiang; Qin, Tong; Xiu, Dongbin SIAM Journal on Scientific Computing, Vol. 42, Issue 6 https://doi.org/10.1137/19M1264011	journal	January 2020
Analysis of Fluid Flows via Spectral Properties of the Koopman Operator Mezić, Igor Annual Review of Fluid Mechanics, Vol. 45, Issue 1 https://doi.org/10.1146/annurev-fluid-011212-140652	journal	January 2013
Learning Reduced Systems via deep Neural Networks with Memory Fu, Xiaohan; Chang, Lo-Bin; Xiu, Dongbin Journal of Machine Learning for Modeling and Computing, Vol. 1, Issue 2 https://doi.org/10.1615/.2020034232	journal	January 2020
Deep Learning of Parameterized Equations with Applications to Uncertainty Quantification Qin, Tong; Chen, Zhen; Jakeman, John D. International Journal for Uncertainty Quantification, Vol. 11, Issue 2 https://doi.org/10.1615/Int.J.UncertaintyQuantification.2020034123	journal	January 2021
Equation-Free, Coarse-Grained Multiscale Computation: Enabling Mocroscopic Simulators to Perform System-Level Analysis Gear, C. William; Hyman, James M.; Kevrekidid, Panagiotis G. Communications in Mathematical Sciences, Vol. 1, Issue 4 https://doi.org/10.4310/CMS.2003.v1.n4.a5	journal	January 2003

Similar Records

Data-Driven Learning of Non-Autonomous Systems

Technical Report · Mon Jun 01 00:00:00 EDT 2020 · OSTI ID:1763550

Data-driven models of nonautonomous systems

Journal Article · Fri Mar 29 00:00:00 EDT 2024 · Journal of Computational Physics · OSTI ID:2350952

Error modeling for surrogates of dynamical systems using machine learning: Machine-learning-based error model for surrogates of dynamical systems

Journal Article · Fri Jul 14 00:00:00 EDT 2017 · International Journal for Numerical Methods in Engineering · OSTI ID:1399882

Related Subjects

97 MATHEMATICS AND COMPUTING
deep neural network
equation recovery
nonautonomous systems
residual network

Data-Driven Learning of Nonautonomous Systems

Citation Formats

References (19)

Similar Records

Related Subjects