Accelerating high-order energy-stable discontinous Galerkin solver using auto-differentiation and neural networks

TL;DR

This paper introduces a differentiable discontinuous Galerkin solver coupled with neural networks that learn correction terms, enabling high-order accurate simulations with reduced computational costs and improved stability through interactive training.

Contribution

It presents a novel differentiable DG solver with neural network corrections, allowing gradient-based optimization and interactive training to enhance accuracy and efficiency in high-order flow simulations.

Findings

NN-corrected low-order simulations match high-order accuracy

Interactive training improves long-term stability and precision

Achieves high-order accuracy with significantly reduced computational cost

Abstract

High-order Discontinuous Galerkin Spectral Element Methods (DGSEM) provide excellent accuracy for complex flow simulations, but their computational cost increases sharply with higher polynomial orders. %that provide very accurate solutions. To alleviate these limitations, this work presents a differentiable DG solver coupled with neural networks (NNs) that learn corrective forcing terms to correct low-order simulations and provide high-order accuracy. The solver's full differentiability enables gradient-based optimization and interactive (solver-in-the-loop) training, mitigating the data-shift problem typically encountered in static, offline learning. Two representative test cases are considered: the one-dimensional viscous Burgers' equation and two-dimensional decaying homogeneous isotropic turbulence (DHIT). The results demonstrate that interactive training with extended unrolling horizons substantially improves the precision and long-term stability of the simulation compared to static training. For the Burgers' equation, a $\mathbb{P}_2$ simulation corrected using a NN-correction achieves the accuracy of a $\mathbb{P}_4$ solution with eight times reduction in computational cost. For the DHIT case, the NN-corrected low-order simulations successfully achieve high-order accuracy while reduce the error beyond the training interval. These results highlight the potential of differentiable solvers combined with neural networks as a robust and efficient framework for accelerating high-fidelity DG-based fluid simulations.