Machine-Learning-Based Modeling of Multiscale Dynamic Systems with Non-Markovian State-Dependent Memory

多尺度动力系统的精确建模一直是一个长期存在的问题，它在计算数学和流体物理、化学工程和材料科学中的应用引起了极大的兴趣。本项目的研究重点是纳米尺度多物理场系统等没有明确尺度分离的多尺度系统。对于这样的系统，现有的方法具有所谓的记忆效应和非马尔可夫行为，这限制了它们的理解和控制。该项目将开发基于机器学习的新型先进计算工具，直接从基于第一性原理的描述中构建多尺度系统的高精度模型。所构建的模型保持了分子水平的保真度，可以广泛应用于研究与材料设计、药物传递和软物质组装相关的动态过程。该项目还将为研究生和本科生提供跨学科的培训和研究经验。本课题的研究将解决无明显尺度分离的动态系统模型约简和多尺度建模的基本问题。由于对状态相关的马尔可夫记忆项的过度简化，当前的经验模型通常显示出保留微尺度水平保真度的局限性，这些记忆项源于微尺度上未解决的动态。这个差距可以通过这个项目中开发的基于机器学习的方法来弥补；这些将提供一个从完整描述中学习一组非马尔可夫特征的一般框架，同时，通过在非马尔可夫特征的扩展动态中嵌入状态相关的记忆项来训练随机约简模型。与传统的机器学习动态方程建模方法不同，本项目中开发的模型基于严格的投影形式化，并保留了清晰的物理解释。简化后的模型可以自然地引入一致噪声项，非常适合于研究复杂的非平衡动态系统。因此，这些方法可以用于研究开放的科学问题，如通过忠实地计算微观尺度的相互作用来研究纳米尺度的组装过程。从长远来看，该项目将为建立此类系统的预测建模和控制提供更全面的计算工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Accurate modeling of multiscale dynamic systems has been a long-standing problem that has attracted tremendous interest in computational mathematics and applications in fluid physics, chemical engineering, and materials science. The focus of this project is on multiscale systems that do not have clear scale separation, such as nanoscale multi-physics systems. For such systems the existing approaches feature so-called memory effects and non-Markovian behavior, which limit their understanding and control. The project will develop new advanced computational tools based on machine learning to construct highly accurate models of multiscale systems directly from first-principle-based descriptions. The constructed models retain a molecular-level fidelity and can be broadly applied to investigate the dynamic processes relevant to material design, drug delivery, and soft matter assembly. This project will also provide interdisciplinary training and research experiences for both graduate and undergraduate students. The research in this project will address a fundamental problem in model reduction and multiscale modeling for dynamic systems without clear scale separation. Current empirical models generally show limitations to retain the microscale level fidelity due to the over-simplification of the state-dependent Markovian memory term arising from the unresolved dynamics on microscale. This gap can be bridged by the methods based on machine-learning developed in this project; these will provide a general framework to learn a set of non-Markovian features from the full descriptions, and simultaneously, train the stochastic reduced model by embedding the state-dependent memory term in the extended dynamics of the non-Markovian features. Different from the conventional machine learning approaches for modeling dynamic equations, the models developed in this project are based on rigorous projection formalism and retain a clear physical interpretation. Consistent noise terms can be naturally introduced to the reduced models, which are well-suited for studying complex dynamic systems out of equilibrium. As a result, the methods can be employed to investigate open scientific questions such as the nanoscale assembly process by faithfully accounting for the microscale interactions. In the long term, this project will provide more comprehensive computational tools for establishing predictive modeling and control of such systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.