2025/12/09 by Xukun Wang, Wang, Xukun, Oscar A. Mariño +3
Engineering · Mathematics · Physics and Astronomy · #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Lattice Boltzmann Simulation Studies #Model Reduction and Neural Networks #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2512.09128
openalex publication_date 2025/12/09 · openalex created_date 2025/12/12 · openalex updated_date 2026/07/28
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 ℙ2 simulation corrected using a NN-correction achieves the accuracy of a ℙ4 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.