Descriptive Neural Stability Controller for Nonlinear PDE Solvers

Abstract

The difficulty of contemporary numerical techniques for solving nonlinear partial differential equations (PDEs) lies in balancing stability and accuracy. To dynamically adjust stability requirements (such as energy decay laws and the Courant–Friedrichs–Lewy (CFL) condition) during equation simulations, this research proposes a metastable neural controller that employs neural networks. To adapt solution parameters (e.g., time step size and pseudo-viscosity) in real time, a neural controller is integrated into explicit time-stepping schemes. To ensure stability and monotonically reduce a specific energy functional, the controller is trained using Lyapunov-inspired losses. We demonstrate the effectiveness of the proposed method on both the Allen–Cahn equation and the viscous Burgers equation. The results show that the neural controller prevents numerical blow-up and preserves the qualitative property of the solution by remaining stable even when time increments exceed those permitted under conventional CFL constraints. The controller’s meta-property yields stable, efficient simulations by inferring stability constraints from the equation state (e.g., derivatives and energy). This study opens new avenues for reliable neural PDE solvers by integrating classical stability theory with data-driven learning.