Junjie Yao , Yuxiao Yi , Liangkai Hang , Weinan E , Weizong Wang , Yaoyu Zhang , Tianhan Zhang , Zhi-Qin John Xu
{"title":"Solving multiscale dynamical systems by deep learning","authors":"Junjie Yao , Yuxiao Yi , Liangkai Hang , Weinan E , Weizong Wang , Yaoyu Zhang , Tianhan Zhang , Zhi-Qin John Xu","doi":"10.1016/j.cpc.2025.109802","DOIUrl":null,"url":null,"abstract":"<div><div>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 <strong>DeePODE</strong> method, which consists of an Evolutionary Monte Carlo Sampling method (EMCS) and an efficient end-to-end deep neural network (DNN) to predict multiscale dynamical systems. The method's primary contribution is its approach to the “curse of dimensionality”– the exponential increase in data requirements as dimensions increase. By integrating Monte Carlo sampling with the system's inherent evolutionary dynamics, DeePODE efficiently generates high-dimensional time-series data covering trajectories with wide characteristic timescales or frequency spectra in the phase space. Appropriate coverage on the frequency spectrum of the training data proves critical for data-driven time-series prediction ability, as neural networks exhibit an intrinsic learning pattern of progressively capturing features from low to high frequencies. We validate this finding across dynamical systems from ecological systems to reactive flows, including a predator-prey model, a power system oscillation, a battery electrolyte thermal runaway, and turbulent reaction-diffusion systems with complex chemical kinetics. The method demonstrates robust generalization capabilities, allowing pre-trained DNN models to accurately predict the behavior in previously unseen scenarios, largely due to the delicately constructed dataset. While theoretical guarantees remain to be established, empirical evidence shows that DeePODE achieves the accuracy of implicit numerical schemes while maintaining the computational efficiency of explicit schemes. This work underscores the crucial relationship between training data distribution and neural network generalization performance. This work demonstrates the potential of deep learning approaches in modeling complex dynamical systems across scientific and engineering domains.</div></div>","PeriodicalId":285,"journal":{"name":"Computer Physics Communications","volume":"316 ","pages":"Article 109802"},"PeriodicalIF":3.4000,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computer Physics Communications","FirstCategoryId":"101","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0010465525003042","RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
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 an Evolutionary Monte Carlo Sampling method (EMCS) and an efficient end-to-end deep neural network (DNN) to predict multiscale dynamical systems. The method's primary contribution is its approach to the “curse of dimensionality”– the exponential increase in data requirements as dimensions increase. By integrating Monte Carlo sampling with the system's inherent evolutionary dynamics, DeePODE efficiently generates high-dimensional time-series data covering trajectories with wide characteristic timescales or frequency spectra in the phase space. Appropriate coverage on the frequency spectrum of the training data proves critical for data-driven time-series prediction ability, as neural networks exhibit an intrinsic learning pattern of progressively capturing features from low to high frequencies. We validate this finding across dynamical systems from ecological systems to reactive flows, including a predator-prey model, a power system oscillation, a battery electrolyte thermal runaway, and turbulent reaction-diffusion systems with complex chemical kinetics. The method demonstrates robust generalization capabilities, allowing pre-trained DNN models to accurately predict the behavior in previously unseen scenarios, largely due to the delicately constructed dataset. While theoretical guarantees remain to be established, empirical evidence shows that DeePODE achieves the accuracy of implicit numerical schemes while maintaining the computational efficiency of explicit schemes. This work underscores the crucial relationship between training data distribution and neural network generalization performance. This work demonstrates the potential of deep learning approaches in modeling complex dynamical systems across scientific and engineering domains.
期刊介绍:
The focus of CPC is on contemporary computational methods and techniques and their implementation, the effectiveness of which will normally be evidenced by the author(s) within the context of a substantive problem in physics. Within this setting CPC publishes two types of paper.
Computer Programs in Physics (CPiP)
These papers describe significant computer programs to be archived in the CPC Program Library which is held in the Mendeley Data repository. The submitted software must be covered by an approved open source licence. Papers and associated computer programs that address a problem of contemporary interest in physics that cannot be solved by current software are particularly encouraged.
Computational Physics Papers (CP)
These are research papers in, but are not limited to, the following themes across computational physics and related disciplines.
mathematical and numerical methods and algorithms;
computational models including those associated with the design, control and analysis of experiments; and
algebraic computation.
Each will normally include software implementation and performance details. The software implementation should, ideally, be available via GitHub, Zenodo or an institutional repository.In addition, research papers on the impact of advanced computer architecture and special purpose computers on computing in the physical sciences and software topics related to, and of importance in, the physical sciences may be considered.