AIMing: Automated Discovery for Structure-Preserving Discretizations

Partial differential equations (PDEs) are one of the most common and flexible mathematical models for physical phenomena, from thermal conduction and fluid dynamics, to waves and electromagnetism. Accordingly, finding approximate solutions to PDEs given physical parameters, such as the geometry of the domain, and the initial and boundary conditions, is of paramount importance in computational and applied mathematics. Defining a discretization of a PDE involves numerous choices, which have a significant impact on accuracy and computational performance. A central idea in numerical analysis is that the discrete problem should be designed to reflect relevant mathematical structures from the continuous problem. This project aims to automate the discovery of structure-preserving discretizations across different kinds of PDEs and different discretization methods. Solvers for partial differential equations play a fundamental role across science and engineering for simulating physical phenomena. The proposed work has the potential to make these important simulations more accurate and more efficient, by providing automated assistance to capture relevant mathematical structure in the discretization, and to discover promising new discretization methods. The proposed work will also develop tools to make known structure-preserving discretizations easier to use and more broadly applicable. The project will involve graduate students. This project is organized along three primary tasks, each tackling a different aspect of discovering structure-preserving discretizations. Our first task aims to find discretizations satisfying conservation properties, one of the most common forms of structure in PDEs describing physical systems. Our second task focuses on finite difference (FD) methods. We will consider automated discovery for FD discretizations that preserve two kinds of structure: positivity, and causality. Our final task turns to finite element methods (FEM). Rather than building on a fixed collection of discrete function spaces studied in the FEM literature, the focus of this task will be to automatically discover suitable shape functions, and compatible collections of function spaces that satisfy mimetic properties. On the more theoretical side, this project has the potential to automatically find new instances of structure-preserving methods for PDEs, which may be of mathematical interest. On the more applied side, the new methods for verification and search developed by this project can lead to new automated tools that make structure-preserving discretizations easier to use in existing frameworks for computational simulation, such as Devito (FD), Fenics, and Firedrake (FEM). 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.