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

TL;DR

This work presents an end-to-end differentiable DGSEM framework that augments a low-order solver with a neural-network–driven corrective forcing to approximate high-order dynamics. By enabling solver-in-the-loop interactive training with extended unrolling horizons, the approach mitigates data-shift and stabilizes long-time integration, achieving high-order accuracy at a fraction of the cost. The method is demonstrated on 1D Burgers' equation and 2D decaying DHIT, showing that NN-corrected low-order simulations can match higher-order results while reducing computational expense, with interactive training outperforming static training in both accuracy and robustness. The results highlight the potential of integrating differentiable solvers with neural corrections to accelerate high-fidelity DG simulations in CFD, while noting challenges in ground-truth data cost, energy stability, and NN design.

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.

Figures (23)

• Figure 1: Schematic diagram of methodology on accelerating high-order simulation using neural network (a) high-order solution trajectory on $\mathcal{M}_{ho}$; (b) trajectories of low-order simulation, training and inference process on $\mathcal{M}_{lo}$ for $n_{\text{unroll}}=1$ (static training); (c) trajectories of training and inference process on $\mathcal{M}_{lo}$ for $n_{\text{unroll}}=4$ (interactive training).
• Figure 2: $\Delta t$ for simulations of different polynomial order $p$.
• Figure 3: Convergence histories of training and validation loss for two MLPs.
• Figure 4: $x-t$ contours of filtered high-order solution $\bm{\bar{q}}_{ho}$, low-order solution $\bm{q}_{lo}$ and low-order solution with NN correction $\bm{q}_{nn}$.
• Figure 5: $x-t$ contours of $l_1$ error of low-order solution $\bm{q}_{lo}$, low-order solution with NN correction $\bm{q}_{nn}$($n_{\text{unroll}}=1$ and $n_{\text{unroll}}=5$) compared to filtered high-order solution $\bm{\bar{q}}_{ho}$.