A robust and stable hybrid neural network/finite element method for 2D flows that generalizes to different geometries

TL;DR

This work advances a robust, hybrid neural network–multigrid solver (DNN-MG) for the 2D incompressible Navier–Stokes equations, coupling a coarse finite element solver with neural corrections on finer grids. Key contributions include stability enhancements via replay buffers, exploration of MLP, RNN, and Transformer architectures, and probabilistic training to model correction uncertainty on unstructured meshes. The authors demonstrate improved generalization across geometries and Reynolds numbers, with significantly reduced time-local errors and practical GPU-enabled speedups, while identifying limitations in highly sensitive or symmetric flow regimes. The approach offers a scalable path to efficient, accurate, and geometry-generalizable fluid simulations, with potential extensions to transport-dominated and probabilistic forecasting tasks.

Abstract

The deep neural network multigrid solver (DNN-MG) combines a coarse-grid finite element simulation with a deep neural network that corrects the solution on finer grid levels, thereby improving the computational efficiency. In this work, we discuss various design choices for the DNN-MG method and demonstrate significant improvements in accuracy and generalizability when applied to the solution of the instationary Navier-Stokes equations. We investigate the stability of the hybrid simulation and show how the neural networks can be made more robust with the help of replay buffers. By retraining on data derived from the hybrid simulation, the error caused by the neural network over multiple time-steps can be minimized without the need for a differentiable numerical solver. Furthermore, we compare multiple neural network architectures, including recurrent neural networks and Transformers, and study their ability to utilize more information from an increased temporal and spatial receptive field. Transformers allow us to make use of information from cells outside the predicted patch even with unstructured meshes while maintaining the locality of our approach. This can further improve the accuracy of DNN-MG without a significant impact on performance.

Figures (15)

• Figure 1: Comparison of the single obstacle case with reference simulations refinement levels 5 and 6 and the NN enhanced simulation using a small MLP to correct the solution on level 5. In all cases, the solution at level 6 is used until $t=0.5$. The results show that the DNN-MG method can develop instabilities that manifest as nonphysical high-frequency features in the flow and that, by some metrics, make the solution worse than the level 5 baseline.
• Figure 2: Design choices regarding patches, which are the basic units for the prediction. For increased clarity, the indicated degrees of freedom (nodes) are for first order elements although we always use quadratic elements in our numerical experiments. The jump level (a) determines the number of mesh subdivisions done starting from the coarse grid for the numerical solution to reach the final degrees of freedom. A patch then corresponds to a cell of the fine mesh. As shown in panel (b), the patch size can be increased by combining cells, reversing the subdivision, while maintaining the degrees of freedom from the fine mesh. Another way to enlarge the receptive field of the neural network is to include multiple input patches from a local neighborhood, as shown in panel (c). Unlike larger patches, this extension does not need to conform to the hierarchical mesh structure, but it requires a suitable neural network architecture to deal with the variable number of inputs. The addition of patches from previous time-steps (d), on the other hand, is always possible.
• Figure 3: Geometry of two example cases (ro1, so4). Each channel has a parabolic inflow profile $\Gamma_{\textrm{in}}$, do-nothing boundary conditions at the outflow boundary $\Gamma_{\textrm{out}}$ and no-slip conditions on the walls $\Gamma_{\textrm{wall}}$ and the obstacles. The thin lines indicate the base mesh which is refined uniformly 5 times for round obstacles and 4 times for square obstacles to create the coarse grid for the numerical simulations.
• Figure 4: The divergence $J_\text{div}$\ref{['eq:div']} and time-averaged velocity error $e_{\bar{v}}$\ref{['eq:mean-vel-error']} of NNs trained on noise augmented data. During training data generation the numerical solution is perturbed by adding noise sampled from a normal distribution with mean $\mu=0$ and standard deviation $\sigma$ to the velocity components with a probability of $c^{-1}$ per time-step ($c=1$ means every time-step). For reference, the mean magnitude of velocity components in the training set without perturbations is $\approx0.95$. Each point represents the mean from 25 NNs trained on the same dataset and the whiskers show the standard deviation.
• Figure 5: Functionals of the single obstacle case ro1 in the standard orientation and with the whole domain rotated by 90 for simulations with different MLPs. Each line represents one MLP. The reference, taken from \ref{['sec:noise-aug']}, is compared to MLPs trained on the same dataset but with rotation augmentation. Small networks have the same size as the reference (1.3d+5 weights), large networks are roughly 8 times larger (1.0d+6 weights).