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

用于振荡纳维-斯托克斯流和非线性算子以及高维福克-普朗克方程的深度神经网络机器学习

• 批准号: 2207449
• 负责人: Wei Cai
• 金额: $ 34.88万
• 依托单位: Southern Methodist University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2207449&HistoricalAwards=false
• 关键词: Deep; Neural; Network; Machine; Learning

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.

这个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已经在语音识别、自动驾驶、图像分类等人工智能(AI)应用中展示了其巨大的能力。这项研究将试图通过提高DNN在学习和理解复杂现象(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)方面的计算和模拟能力，来利用DNN的能力进行科学发现。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，具有更好的频率分辨率来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，并增加从复杂生化系统的过渡路径理论(TPT)求解高维Fokker-Planck方程(FPE)时解决维度灾难(COD)的能力。具体地说，将在与科学和工程计算有关的三个主要计算问题上开展研究：(A)开发多尺度DNN作为一种可行的无网格方法来求解复杂区域中依赖时间的高度振荡的Navier-Stokes流，以及一种实用的替代传统数值方法的方法，而不需要昂贵的网格剖分或线性系统求解器。(B)开发用于地震波预测、正反向散射应用的高振荡函数空间中的非线性算子的多尺度DNN学习算法。(C)发展基于向前和向后随机微分方程(FBSDE)的多尺度DNN，用于高维偏微分方程组边值问题，例如由物理系统的统计描述产生的Fokker-Planck方程，并应用于计算复杂化学和生物系统的跃迁路径抽样理论中的提交函数和跃迁率。这一研究将对提高ML的科学计算能力产生广泛的影响。对于复杂几何结构中的振荡流动和具有高度振荡性质的物理性质的非线性算子的表示，高效的无网格DNN算法可以减少许多优化以及科学和工程研究中的正问题和反问题的成本。用多尺度FBSDE DNN算法解决高维Fokker-Planck方程的离散问题，将对ML的研究以及材料科学与最优控制、金融工程、生物系统动力学等多个科学领域产生广泛的影响。此外，该项目将高度重视数据科学教育，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智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

FBSDE based neural network algorithms for high-dimensional quasilinear parabolic PDEs

基于 FBSDE 的高维拟线性抛物线偏微分方程神经网络算法

• DOI: 10.1016/j.jcp.2022.111557
• 发表时间: 2022
• 期刊: Journal of Computational Physics
• 影响因子: 4.1
• 作者: Zhang, Wenzhong;Cai, Wei
• 通讯作者: Cai, Wei

Linearized Learning with Multiscale Deep Neural Networks for Stationary Navier-Stokes Equations with Oscillatory Solutions

使用多尺度深度神经网络进行线性学习，用于具有振荡解的稳态纳维-斯托克斯方程

• DOI: 10.4208/eajam.2022-328.230423
• 发表时间: 2022
• 期刊: East Asian Journal on Applied Mathematics
• 影响因子: 1.2
• 作者: Liu, Lizuo;null, Bo Wang;Cai, Wei
• 通讯作者: Cai, Wei