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

玻尔兹曼方程在从外部空气动力学和推进器羽流到真空设施和微尺度设备的广泛应用中出现。波尔兹曼方程的新兴应用包括自组织系统和集群，以及网络和细菌动力学。这些领域的持续技术进步要求改进算法和模型，以便能够为手头的问题开发一个“数字孪生兄弟”。尽管玻尔兹曼方程为这些系统提供了最准确的模型，但由于其高昂的计算成本，它在多个空间维度中的应用仍然有限。这个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来使Boltzmann方程的确定解变得容易处理，从而可以应用于新的工程应用的模拟。该项目将支持加州州立大学北岭分校的新课程和新的培训机会，该大学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其高维度和评估五重碰撞积分的高昂成本。为了解决这些问题，这个项目将专注于数据驱动的波尔兹曼方程低维离散的开发、实施和验证。该项目将开发方法来加强降阶模型的长期稳定性，并设计基于解数据的稳定的宏观模型。利用深度残差神经网络和数值梯度流方法建立动力学方程的快速模型。此外，还将研究八叉树网格上卷积的有效计算和Boltzmann方程的最优输运公式。求解器将使用可用的确定性高阶精确求解器进行验证。该项目将提供非连续流三维解的算法、基准解以及开发和测试气体近似动力学和宏观模型的新方法。这些技术将扩大非连续统解算器的适用范围，并将在以前昂贵的模型中包含多种物理因素。新方法将适用于大气再入、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。这一奖项反映了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。

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.