Deep Neural Network Machine Learning for Oscillatory Navier-Stokes Flows and Nonlinear Operators, and High Dimensional Fokker-Planck Equations

该项目的目标是开发具有深度神经网络（DNN）的机器学习（ML）技术，以研究广泛的物理和化学现象。DNN已经在人工智能（AI）应用中展示了其巨大的力量，如语音识别，自动驾驶，图像分类等，这项研究将试图通过增加学习和理解复杂现象的计算和模拟能力来利用DNN的力量进行科学发现，如造船应用中的流体流动，地震波预测，药物设计中的化学反应。在这个项目中，为了解决科学计算中的一些关键计算挑战，将开发新的深度神经网络（DNN）ML算法，这些算法具有更好的频率分辨率，用于模拟复杂几何形状中的振荡流体流动和振荡函数空间中的非线性算子，并增加了从复杂生化系统的过渡路径理论（TPT）求解高维Fokker-Planck方程（FPE）中解决维数灾难（CoD）的能力。具体而言，将在与科学和工程计算相关的三个主要计算问题上进行研究：（a）开发多尺度DNN作为一种可行的无网格方法，用于解决复杂区域中与时间相关的高振荡Navier-Stokes流，以及一种实用的替代方法，以取代传统的数值方法，无需昂贵的网格生成或线性系统求解器。(b)开发用于地震波预测以及正向和反向散射应用的高度振荡函数空间中的非线性算子的多尺度DNN学习算法。(c)开发基于前向和后向随机微分方程（FBNN）的多尺度DNN，用于高维偏微分方程（如Fokker-Planck方程）的边值问题，这些边值问题源于物理系统的统计描述，并应用于计算复杂化学和生物系统过渡路径采样理论中的提交函数和过渡率。这项研究将在许多方面对提高ML的科学计算能力产生广泛的影响。用无网格DNN算法求解复杂几何条件下的振荡流，用非线性算子表示高振荡性质的物理量，可以减少科学和工程研究中许多优化问题、正问题和反问题的求解成本。用多尺度FBNN算法解决高维Fokker-Planck方程的CoD问题将对ML研究和许多科学领域产生广泛的影响，如材料科学和最优控制，金融工程，以及生物系统动力学。此外，该项目将重点关注数据科学教育，在SMU数据科学研究所开展新的ML课程开发和活动。该奖项反映了NSF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。

The goal of this project is to develop machine learning (ML) technology with deep neural networks (DNNs) to study a wide range of physical and chemical phenomena. DNN has demonstrated its tremendous power in artificial intelligence (AI) applications such as speech recognition, automatic self-driving, image classification, etc. This research will attempt to harness the power of the DNN for scientific discoveries by increasing computational and simulation capabilities in learning and understanding complex phenomena such as fluid flows for ship building applications, seismic wave predictions, chemical reactions in drug designs. In this project, to address some of the key computational challenges in scientific computing, new classes of deep neural network (DNN) ML algorithms will be developed with capabilities for better frequency resolution for simulating oscillatory fluid flows in complex geometries and nonlinear operators in oscillatory function spaces, and increased powers for addressing the curse of dimensionality (CoD) in solving high dimensional Fokker-Planck equations (FPE) from transition path theory (TPT) of complex biochemical systems. Specifically, research will be carried out in three major computational issues relevant to scientific and engineering computing: (a) To develop multiscale DNNs as a viable meshless method for solving time dependent highly oscillatory Navier-Stokes flows in complex domains, and a practical alternative method to traditional numerical methods with no costly mesh generation or linear system solvers. (b) To develop multiscale DNN learning algorithms for nonlinear operators in highly oscillatory function spaces for seismic wave prediction, and forward and inverse scattering applications. (c) To develop forward and backward stochastic differential equation (FBSDE) based multiscale DNNs for boundary value problems of high dimensional PDEs such as Fokker-Planck equations arising from statistical description of physical systems with application in computing committor functions and transition rates in transition path sampling theory of complex chemical and biological systems. This research will have a broad impact in improving the capability of ML for scientific computing in many ways. Efficient meshless DNN algorithms for oscillatory flows in complex geometry and representation of nonlinear operator for physical qualities of highly oscillatory nature can reduce the cost of many optimization, and forward and inverse problems in scientific and engineering research. Overcoming the CoD in solving high dimensional Fokker-Planck equations with the multi-scale FBSDE DNN algorithms will have a wide range impact on both the ML research and many scientific areas such as material sciences and optimal control, financial engineering, as well as to biological system dynamics. Moreover, the project will place strong emphasis on data science education with new ML course development and activities in the Data Science Institute at SMU.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.