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

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.