Learning Effective Dynamics across Spatio-Temporal Scales of Complex Flows

TL;DR

The paper tackles the challenge of simulating multiscale spatiotemporal dynamics in incompressible flows on unstructured meshes, where full-resolution CFD is too expensive. It introduces Graph-LED, which combines a graph neural network (GNN) encoder–decoder for spatial dimension reduction with a Transformer-based autoregressive temporal model to forecast reduced dynamics. The method maps the high-dimensional state $U_0$ to a latent $Z_0$, propagates for $n$ steps via the Transformer, and decodes back to $U_n$; training is decoupled for efficiency. Evaluations on flow past a cylinder at $Re=696$ and flow over a backward-facing step at $Re=5000$ show close agreement with high-fidelity simulations while delivering up to $900$-fold speedups, outperforming existing graph-based baselines on key metrics.

Abstract

Modeling and simulation of complex fluid flows with dynamics that span multiple spatio-temporal scales is a fundamental challenge in many scientific and engineering domains. Full-scale resolving simulations for systems such as highly turbulent flows are not feasible in the foreseeable future, and reduced-order models must capture dynamics that involve interactions across scales. In the present work, we propose a novel framework, Graph-based Learning of Effective Dynamics (Graph-LED), that leverages graph neural networks (GNNs), as well as an attention-based autoregressive model, to extract the effective dynamics from a small amount of simulation data. GNNs represent flow fields on unstructured meshes as graphs and effectively handle complex geometries and non-uniform grids. The proposed method combines a GNN based, dimensionality reduction for variable-size unstructured meshes with an autoregressive temporal attention model that can learn temporal dependencies automatically. We evaluated the proposed approach on a suite of fluid dynamics problems, including flow past a cylinder and flow over a backward-facing step over a range of Reynolds numbers. The results demonstrate robust and effective forecasting of spatio-temporal physics; in the case of the flow past a cylinder, both small-scale effects that occur close to the cylinder as well as its wake are accurately captured.

Figures (11)

• Figure 1: Overview of Graph-LED: A high-dimensional initial state $\boldsymbol{U}_0$ is mapped to a low-dimensional latent representation $\boldsymbol{Z}_0$ via a GNN Encoder. Subsequently this latent representation is propagated for $n$ steps using a Transformer. The GNN Decoder then maps the forecasted low-dimensional state $\boldsymbol{Z}_n$ back to the high-dimensional state of interest $\boldsymbol{U}_n$.
• Figure 2: Left: Visualization of a FV mesh that can be naturally represented by a Graph. Right: Definition of graph nodes $\boldsymbol{u}_j$ as cells and graph edges $\boldsymbol{e}_ij$ as connection between two adjacent cells
• Figure 3: Interpolation to new coordinates: Based on the three colored nodes, we reconstruct the node values at new nodes via nearest neighbor interpolation, i.e. weighting the contribution of each colored node according to their distance from the new node. Finally we are able to construct a new larger or smaller graph.
• Figure 4: The system is reduced from 27,127 points to 1,024 points.
• Figure 5: Vorticity ($\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$) forecasted by Graph-LED (left), OpenFOAM (middle) and the error (right) from $t = 0, 50, 100$ (from top row to bottom row).

Theorems & Definitions (9)

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