New Computational Framework for Complex Kinetic Simulation and Prediction

Kinetic theory has transformed our understanding of interacting particle systems in both nature and engineering. Despite their importance, kinetic equations remain challenging to solve. This difficulty arises from their high dimensionality, the presence of multiple scales, and the need to preserve key structures such as conservation, positivity, and entropy dissipation. Additionally, the multi-query task of parameter identification places a higher demand on solver's efficiency. This project intends to address these challenges by developing efficient and scalable variational computational methods. These methods will integrate ideas from optimal transport, scientific machine learning, and stochastic methods, along with the unique structure of kinetic equations. The project also includes the training of graduate students, contributing to the development of the next generation of computational mathematicians. The project has two main objectives. The first is to develop and analyze learning-enhanced, structure-preserving particle methods for nonlinear partial differential equations, with a particular focus on plasma models. The methods will preserve both the Hamiltonian structure of the field terms and the dissipative nature of the collision. They are intended to complement existing particle-in-cell approaches for collisionless plasmas and to offer improvements in scalability and stability for collisional plasma simulations. The second objective is to design reduced-order methods for optimization problems constrained by kinetic equations. This will involve leveraging the multiscale nature of the equations or employing intelligent use of randomness. The proposed methods aim to meet the pressing need for efficient inverse solvers, particularly given the growing applications of kinetic theory to real-world problems. 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.