The modeling and simulation of high-dimensional multiscale systems is a critical challenge across all areas of science and engineering. It is broadly believed that even with today's computer advances resolving all spatiotemporal scales described by the governing equations remains a remote target. This realization has prompted intense efforts to develop model order reduction techniques. In recent years, techniques based on deep recurrent neural networks have produced promising results for the modeling and simulation of complex spatiotemporal systems and offer large flexibility in model development as they can incorporate experimental and computational data. However, neural networks lack interpretability, which limits their utility and generalizability across complex systems. Here we propose a novel framework of Interpretable Learning Effective Dynamics (iLED) that offers comparable accuracy to state-of-the-art recurrent neural network-based approaches while providing the added benefit of interpretability. The iLED framework is motivated by Mori-Zwanzig and Koopman operator theory, which justifies the choice of the specific architecture. We demonstrate the effectiveness of the proposed framework in simulations of three benchmark multiscale systems. Our results show that the iLED framework can generate accurate predictions and obtain interpretable dynamics, making it a promising approach for solving high-dimensional multiscale systems.

翻译：高维多尺度系统的建模与模拟是横跨所有科学与工程领域的重大挑战。普遍认为，即便借助当今最先进的计算机技术，完全解析控制方程所描述的所有时空尺度仍是一个遥远的目标。这一认识推动了模型降阶技术的深入研究。近年来，基于深度循环神经网络的方法在复杂时空系统的建模与模拟中展现了令人瞩目的成果，并因其能够融合实验与计算数据而在模型开发中提供了极大的灵活性。然而，神经网络缺乏可解释性，这限制了其在复杂系统中的实用性与泛化能力。本文提出了一种名为"可解释学习有效动力学"（iLED）的新型框架，该框架在保持与最先进循环神经网络方法相当的精度的同时，额外提供了可解释性的优势。iLED框架的设计灵感源于Mori-Zwanzig理论与Koopman算子理论，这为特定架构的选择提供了理论依据。我们在三个基准多尺度系统的模拟中验证了所提框架的有效性。结果表明，iLED框架既能生成准确的预测，又能获得可解释的动力学特征，使其成为解决高维多尺度系统问题的一种颇具前景的方法。