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.

翻译：多尺度动力系统由具有广泛特征时间尺度的高维刚性常微分方程建模，广泛存在于科学与工程领域，但其数值求解器常面临严峻的效率瓶颈。本文提出新型DeePODE方法，该方法包含全局多尺度采样策略和深度神经网络拟合技术，以处理多尺度系统。DeePODE的核心贡献在于通过融合蒙特卡洛方法与常微分系统内在演化特性，高效发掘代表性训练集，从而解决多尺度挑战，且不陷入"维数灾难"。该DeePODE方法在多个领域的不同多尺度系统中得到验证，包括捕食者-猎物模型、电力系统振荡、电池电解液自燃及湍流火焰。我们的方法对未见工况展现出强大的泛化能力，充分彰显了深度学习在建模科学与工程领域复杂多尺度动力过程中的强大潜力。