Accelerating high-order energy-stable discontinous Galerkin solver using auto-differentiation and neural networks
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909953216741376 |
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| author | Wang, Xukun Marino, Oscar A. Ferrer, Esteban |
| author_facet | Wang, Xukun Marino, Oscar A. Ferrer, Esteban |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09128 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Accelerating high-order energy-stable discontinous Galerkin solver using auto-differentiation and neural networks Wang, Xukun Marino, Oscar A. Ferrer, Esteban Fluid Dynamics 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. |
| title | Accelerating high-order energy-stable discontinous Galerkin solver using auto-differentiation and neural networks |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/2512.09128 |