RUI: Development of Fast Methods for Solving the Boltzmann Equation through Reduced Order Models, Machine Learning, and Optimal Transport

玻尔兹曼方程的应用范围很广，从外部空气动力学和推力器羽流到真空设备和微尺度设备。玻尔兹曼方程的新兴应用包括自组织系统和群集，以及网络和细菌动力学。这些领域的持续技术进步需要改进算法和模型，以便为手头的问题开发“数字双胞胎”。虽然玻尔兹曼方程为这些系统提供了最精确的模型，但由于其高昂的计算成本，它在多个空间维度上的应用仍然受到限制。该项目的目标是利用数据驱动的降阶模型、机器学习和最优传输理论，使玻尔兹曼方程的确定性解易于处理，从而可以应用于新型工程应用的模拟。该项目将支持加州州立大学北岭分校的新课程和新培训机会，这是一所少数族裔服务机构。求解玻尔兹曼方程的主要困难是它的高维数和计算五重碰撞积分的高昂代价。为了解决这些问题，本项目将重点关注玻尔兹曼方程的数据驱动低维离散化的开发、实施和验证。该项目将开发方法来增强降阶模型的长期稳定性，并设计基于解数据的稳定宏观模型。动力学方程的快速模型将使用深度残差神经网络和数值梯度流方法来开发。此外，将研究八叉树网格上卷积的有效评估和玻尔兹曼方程的最优输运公式。求解器将使用现有的确定性高阶精确求解器进行验证。该项目将为非连续流体的三维解决方案、基准解决方案以及开发和测试气体近似动力学和宏观模型的新方法提供算法。这些技术将增加非连续介质求解器的适用范围，并将多个物理模型纳入以前昂贵得令人望而却步的模型中。这些新方法将应用于大气再入、高超声速流动以及气体驱动的芯片实验室技术、微推进和原子力显微镜的模拟。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。

The Boltzmann equation arises in a wide range of applications from external aerodynamics and thruster plume flows to vacuum facilities and microscale devices. Emerging applications of the Boltzmann equation include self-organizing systems and flocking, as well as networks and bacterial dynamics. Continuing technological advances in these areas require improvement of algorithms and models to enable development of a “digital twin” for the problems at hand. While the Boltzmann equation provides the most accurate model for these systems, its use in multiple spatial dimensions remains limited due to its prohibitive computational costs. The goal of this project is to leverage data-driven reduced order models, machine learning, and optimal transport theory to make deterministic solution of the Boltzmann equation tractable so it can be applied to simulation of novel engineering applications. The project will support new courses and new training opportunities at California State University Northridge which is a minority serving institution.The key difficulties in solving the Boltzmann equation are its high dimensionality and the prohibitive costs of evaluating the five-fold collision integral. To address these, this project will focus on the development, implementation, and validation of data driven low dimensional discretizations of the Boltzmann equation. The project will develop methods to enforce long term stability of the reduced order models and to design stable macroscopic models that are based on solution data. Fast models for kinetic equations will be developed using deep residual neural networks and numerical gradient flow approaches. Additionally, efficient evaluation of convolution on octree meshes and optimal transport formulation of the Boltzmann equation will be studied. The solvers will be validated using available deterministic high order accurate solvers. The project will deliver algorithms for three-dimensional solutions of non-continuum flows, benchmark solutions as well as new approaches to develop and test approximate kinetic and macroscopic models of gas. The techniques will increase the range of applicability of non-continuum solvers and will include multiple physics into models that were previously prohibitively expensive. The new methods will apply to the simulations of atmospheric re-entry, hypersonic flows and also gas-driven lab-on-the-chip technologies, micropropulsion, and atomic force microscopy.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.