Machine Learning for Effective Computation in Multiscale Hyperbolic Systems

在物理科学和工程应用中，经常需要求解涉及多个时间尺度和长度尺度的微分方程组。这些方程的直接数值模拟通常在计算上是不可行的。该研究计划旨在通过开发利用机器学习和并行时间算法的最新进展的算法，减少应用程序中发现的多尺度动态系统模拟的总体时钟计算时间。目标是提供一种可以推广到其他多尺度时间相关问题的系统方法，并提高一类现有方法的有效性。该算法将在大规模并行百亿亿次计算机架构上实现包括分子动力学和地震成像在内的一系列重要应用的逼真计算机模拟。这个项目的机器学习方面将吸引学生对科学计算和刺激进一步的跨学科工作。在数据科学和人工智能时代，这样的培训对于培养下一代科学计算研究人员至关重要。该项目涉及多尺度振荡动力系统，其中相位误差通常主导数值解并且不随时间消散，使得精确的长时间模拟非常困难。本研究计划旨在利用现代机器学习方法，结合良好的训练示例，对系统中的物理和因果关系进行适当的采样，构建多尺度动力系统的有效解算子。因此，本研究计划的中心重点是开发一个框架，用于生成有效的训练数据，正确地采样双曲系统中的强因果关系。所研究的基于流的集成采样策略利用了目标系统的物理特性。该程序还包括一个“自我改进”的迭代过程，不仅可以实现大规模并行计算，还可以提高基于机器学习的计算的准确性。该项目将直接涉及两名研究生和本科生的指导。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。

In physical science and engineering applications, it is often necessary to solve systems of differential equations that involve many temporal and length scales. Direct numerical simulations for these equations are often computationally infeasible. This research program aims to reduce the overall wall-clock computation time for simulations of multiscale dynamical systems found in applications by developing algorithms that leverage recent advancements in machine learning and parallel-in-time algorithms. The goal is to provide a systematic approach that can be generalized to other multiscale time-dependent problems and improve the efficacy of a class of existing methodologies. The algorithms will enable realistic computer simulations in a class of important applications, including molecular dynamics and seismic imaging, on massively parallel exascale computer architectures. The machine learning aspects of this project will attract students to scientific computing and stimulate further interdisciplinary work. Such training will be essential for preparing the future generation of researchers in scientific computing in the era of data science and artificial intelligence. This project concerns multiscale oscillatory dynamical systems, in which phase errors typically dominate the numerical solutions and do not dissipate in time, making accurate long-time simulations very difficult. This research program aims to construct effective solution operators for multiscale dynamical systems by using modern machine learning approaches combined with good training examples that properly sample the physics and causality in the system. Accordingly, a central focus of this research program is to develop a framework for the generation of effective training data that correctly sample the strong causality in the hyperbolic systems. The flow-based ensemble sampling strategy under study exploits the physical properties of the targeted systems. The program also includes a “self-improving” iterative procedure, which not only enables massive parallel-in-time computation but also improves the accuracy of machine learning-based computations. The project will directly involve two graduate students and the mentoring of undergraduate students.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.