How To Design Stable Machine Learned Solvers For Scalar Hyperbolic PDEs

Nick McGreivy, Ammar Hakim

Keywords: machine learning for sciences, machine learning for physics, machine learning for numerical methods, numerical methods, partial differential equations

TL;DR: We introduce a novel technique to guarantee the stability of scalar hyperbolic PDE solvers which can be used as an error-correcting algorithm for machine-learned PDE solvers.

Abstract: Machine learned partial differential equation (PDE) solvers trade the robustness of classical numerical methods for potential gains in accuracy and/or speed. A key challenge for machine learned PDE solvers is to maintain physical constraints that will improve robustness while still retaining the flexibility that allows these methods to be accurate. In this paper, we show how to design solvers for scalar hyperbolic PDEs that are stable by construction. We call our technique 'global stabilization.' Unlike classical numerical methods, which guarantee stability by putting local constraints on the solver, global stabilization adjusts the time-derivative of the discrete solution to ensure that global invariants and stability conditions are satisfied. Although global stabilization can be used to ensure the stability of any scalar hyperbolic PDE solver that uses method of lines, it is designed for machine learned solvers. Global stabilization's unique design choices allow it to guarantee stability without degrading the accuracy of an already-accurate machine learned solver.