Solving multiscale dynamical systems by deep learning

Multiscale dynamical systems, modeled by high-dimensional stiff ordinary differential equations (ODEs) with wide-ranging characteristic timescales, arise across diverse fields of science and engineering, but their numerical solvers often encounter severe efficiency bottlenecks. This paper introduces...

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Bibliographic Details
Published in:arXiv.org 2024-01
Main Authors: Zhi-Qin John Xu, Yao, Junjie, Yi, Yuxiao, Liangkai Hang, Weinan, E, Zhang, Yaoyu, Zhang, Tianhan
Format: Article
Language:English
Subjects:
Artificial neural networks
Batteries
Deep learning
Differential equations
Dynamical systems
Machine learning
Monte Carlo simulation
Predator-prey simulation
Spontaneous combustion
Turbulent flames
Online Access:Get full text
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Summary:Multiscale dynamical systems, modeled by high-dimensional stiff ordinary differential equations (ODEs) with wide-ranging characteristic timescales, arise across diverse fields of science and engineering, but their numerical solvers often encounter severe efficiency bottlenecks. This paper introduces a novel DeePODE method, which consists of a global multiscale sampling method and a fitting by deep neural networks to handle multiscale systems. DeePODE's primary contribution is to address the multiscale challenge of efficiently uncovering representative training sets by combining the Monte Carlo method and the ODE system's intrinsic evolution without suffering from the ``curse of dimensionality''. The DeePODE method is validated in multiscale systems from diverse areas, including a predator-prey model, a power system oscillation, a battery electrolyte auto-ignition, and turbulent flames. Our methods exhibit strong generalization capabilities to unseen conditions, highlighting the power of deep learning in modeling intricate multiscale dynamical processes across science and engineering domains.
ISSN:2331-8422