Solving multiscale dynamical systems by deep learning

IF 3.4 2区 物理与天体物理 Q1 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS
Junjie Yao , Yuxiao Yi , Liangkai Hang , Weinan E , Weizong Wang , Yaoyu Zhang , Tianhan Zhang , Zhi-Qin John Xu
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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.
用深度学习求解多尺度动力系统
摘要以具有广泛特征时标的高维刚性常微分方程(ode)为模型的多尺度动力系统,在科学和工程的各个领域都有应用,但其数值求解常常遇到严重的效率瓶颈。提出了一种新的DeePODE方法,该方法由进化蒙特卡罗采样方法(EMCS)和高效的端到端深度神经网络(DNN)组成,用于预测多尺度动力系统。该方法的主要贡献在于它解决了“维数的诅咒”——随着维数的增加,数据需求呈指数级增长。通过将蒙特卡罗采样与系统固有的进化动力学相结合,DeePODE有效地生成高维时间序列数据,这些数据覆盖了相空间中具有宽特征时间尺度或频谱的轨迹。在训练数据的频谱上适当的覆盖对于数据驱动的时间序列预测能力至关重要,因为神经网络表现出从低到高频率逐步捕获特征的内在学习模式。我们在从生态系统到反应流的动力学系统中验证了这一发现,包括捕食者-猎物模型、电力系统振荡、电池电解质热失控以及具有复杂化学动力学的湍流反应-扩散系统。该方法展示了强大的泛化能力,允许预训练的DNN模型在以前未见过的场景中准确预测行为,这主要归功于精心构建的数据集。虽然理论保证仍有待建立,但经验证据表明,DeePODE在保持显式格式的计算效率的同时,实现了隐式数值格式的精度。这项工作强调了训练数据分布与神经网络泛化性能之间的关键关系。这项工作证明了深度学习方法在跨科学和工程领域建模复杂动态系统方面的潜力。
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来源期刊
Computer Physics Communications
Computer Physics Communications 物理-计算机:跨学科应用
CiteScore
12.10
自引率
3.20%
发文量
287
审稿时长
5.3 months
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.
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