Collaborative Research: Theory, computation and applications of parameterized solution operator approximation for high-dimensional PDEs

Partial differential equations (PDEs) play a fundamental role in modeling physical laws, chemical and biological processes, financial systems, and modern engineering designs. Despite their importance, most PDEs do not admit analytical solutions, necessitating the use of numerical simulations. While numerical methods have achieved considerable success over the past decades, solving high-dimensional PDEs and simulating PDE solution operators remain major challenges due to the curse of dimensionality and high computational demands. Recent breakthroughs in deep neural networks (DNNs) have opened new avenues in scientific computing. These developments provide promising tools for addressing difficult problems in applied mathematics. This project aims to develop novel mathematical theories and computation methods to efficiently solve high-dimensional PDEs and to learn solution operators using DNN-based approaches. The research will offer rich opportunities for training the new generation of applied and computational mathematicians and engineers. The project focuses on three interrelated objectives that leverage advanced nonlinear reduced-order models with recent developments in optimal transport theory and operator learning. First, it proposes a supervised learning method for solving high-dimensional Hamilton-Jacobi equations using a density coupling strategy. Second, it develops a parameter control framework to enable rapid simulations of high-dimensional evolution PDEs across varying initial and boundary conditions. Third, it introduces a deep tangent bundle method for efficient high-dimensional function approximation and PDE simulation. These contributions will be accompanied by a rigorous theoretical analysis covering model properties, computational complexity, and error bounds. 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.