Learning Distributions of Complex Fluid Simulations with Diffusion Graph Networks

TL;DR

This work tackles the challenge of obtaining full statistical distributions of unsteady fluid states without long simulations. It introduces diffusion on graphs (DGN) and a latent-space variant (LDGN) to sample converged flow states conditioned on mesh geometry and physical parameters, with a multi-scale GNN processor and a VGAE latent space to improve efficiency and reduce high-frequency artifacts. The methods are validated on Ellipse, EllipseFlow, and Wing domains, showing that LDGN more accurately captures distributions and generalizes across parameter settings while delivering substantial speedups over conventional CFD runs. The approach enables efficient computation of flow statistics (e.g., RMS, two-point correlations) and holds promise for complex, large-scale applications where direct CFD is prohibitive. Future directions include incorporating temporal conditioning and flow matching to further enhance temporal correlations and sampling speed.

Abstract

Physical systems with complex unsteady dynamics, such as fluid flows, are often poorly represented by a single mean solution. For many practical applications, it is crucial to access the full distribution of possible states, from which relevant statistics (e.g., RMS and two-point correlations) can be derived. Here, we propose a graph-based latent diffusion (or alternatively, flow-matching) model that enables direct sampling of states from their equilibrium distribution, given a mesh discretization of the system and its physical parameters. This allows for the efficient computation of flow statistics without running long and expensive numerical simulations. The graph-based structure enables operations on unstructured meshes, which is critical for representing complex geometries with spatially localized high gradients, while latent-space diffusion modeling with a multi-scale GNN allows for efficient learning and inference of entire distributions of solutions. A key finding is that the proposed networks can accurately learn full distributions even when trained on incomplete data from relatively short simulations. We apply this method to a range of fluid dynamics tasks, such as predicting pressure distributions on 3D wing models in turbulent flow, demonstrating both accuracy and computational efficiency in challenging scenarios. The ability to directly sample accurate solutions, and capturing their diversity from short ground-truth simulations, is highly promising for complex scientific modeling tasks.

Figures (19)

• Figure 1: (a) We learn the probability distribution of the systems' converged states provided only a short trajectory of length $\delta << T$ per system. (b) A preview of our turbulent wing experiment. The distribution learned by our LDGN model accurately captures the variance of all states (bottom right), despite seeing only an incomplete distribution for each wing during training (top right).
• Figure 2: (a) Our VGAE consists of a condition encoder, a (node) encoder, and a (node) decoder. The multi-scale latent features from the condition encoder serve as conditioning inputs to both the encoder and the decoder. (b) During LDGN inference, Gaussian noise is sampled in the VGAE latent space and, after multiple denoising steps conditioned on the low-resolution outputs from the VGAE's condition encoder, transformed into the physical space by the VGAE's decoder.
• Figure 3: (L)DGNs can generate diverse states of dynamic mesh-based simulations given the mesh geometry and simulation parameters. We demonstrate this by learning: (a) the pressure on an ellipse in 2D laminar flow (Ellipse task), (b) the velocity and pressure fields resulting from that flow (EllipseFlow task), and (c) the pressure on a wing in 3D turbulent flow (Wing task).
• Figure 4: For a system on dataset Ellipse-InDist, (a) samples from the LDGN and ground truth, and (b) probability density function from the DGN, LDGN, baseline models, and ground truth. The DGN and LDGN show the best distributional accuracy.
• Figure 5: Samples from the DGN, LDGN, baseline models, and ground truth on (a) dataset EllipseFlow-InDist, and (b) dataset Wing-Test. The DGNs and LDGNs achieve the highest sample accuracy, with the DGNs showing good accuracy but retaining some high-frequency noise.