Hybrid ML-Numerical Solvers for Hyperbolic-Parabolic PDEs: Overcoming Timestep Constraints

This work presents a novel hybrid numerical methodology for solving partial differential equations (PDEs) characterized by the co-existence of hyperbolic and parabolic terms. A well-known bottleneck in conventional numerical schemes is the parabolic stability constraint, which mandates severely restricted timesteps or necessitates the adoption of computationally expensive implicit solvers in practical astrophysics simulations. We demonstrate that the strategic integration of a Deep Neural Network (DNN) within a numerical solver provides an efficient means to overcome this restrictive parabolic timestep, thereby enabling the evolution of both hyperbolic and parabolic dynamics at the hyperbolic timestep limit. Our investigation contrasts DNNs with Fourier Neural Operators (FNOs) as candidate neural network architectures, revealing that FNOs exhibit markedly superior stability, attributable to their inherent design for mitigating high-wavenumber errors. We further provide a linear stability analysis to precisely delineate the operational regimes and potential instabilities of our ML-enhanced models. Preliminary studies on the Burgers-Diffusion equation and the viscous Euler's equations with heat conduction serve to validate the efficacy of our proposed framework.