Physics-aware generative models for turbulent fluid flows through energy-consistent stochastic interpolants

TL;DR

This work introduces physics-aware generative modeling for turbulence by embedding energy stability and divergence-freeness into stochastic interpolants. By optimizing interpolant coefficients to enforce energy conservation in expectation and enforcing a hard divergence-free constraint via projection, the method achieves stable long-rollout surrogates for incompressible Navier–Stokes turbulence. On Kolmogorov flow, the energy-consistent SI (SI_opt_div) outperforms DDPM-based baselines (ACDM, PDE-Refiner) in energy distributions and spectral fidelity, while enabling flexible inference with fewer diffusion steps. The approach offers a scalable nonintrusive surrogate that preserves fundamental conservation properties and enhances the reliability of probabilistic turbulence forecasts.

Abstract

Generative models have demonstrated remarkable success in domains such as text, image, and video synthesis. In this work, we explore the application of generative models to fluid dynamics, specifically for turbulence simulation, where classical numerical solvers are computationally expensive. We propose a novel stochastic generative model based on stochastic interpolants, which enables probabilistic forecasting while incorporating physical constraints such as energy stability and divergence-freeness. Unlike conventional stochastic generative models, which are often agnostic to underlying physical laws, our approach embeds energy consistency by making the parameters of the stochastic interpolant learnable coefficients. We evaluate our method on a benchmark turbulence problem - Kolmogorov flow - demonstrating superior accuracy and stability over state-of-the-art alternatives such as autoregressive conditional diffusion models (ACDMs) and PDE-Refiner. Furthermore, we achieve stable results for significantly longer roll-outs than standard stochastic interpolants. Our results highlight the potential of physics-aware generative models in accelerating and enhancing turbulence simulations while preserving fundamental conservation properties.

Figures (13)

• Figure 1: Visualization of the various steps and components in the energy-consistent stochastic interpolant framework. At the top, the existing framework presented in chen_probabilistic_2024 is visualized. The training is performed by sampling two consecutive physical states and interpolating between them via the stochastic interpolant in pseudo-time. The interpolant is used to train a drift term in an SDE that will be solved in pseudo-time during inference to generate new states conditioned on the initial state. Choosing the interpolant without including physics knowledge can result in inconsistencies in energy with respect to the physical energy, as visualized in the lower left part of the figure. In this paper, we propose to optimize the interpolant for energy-consistency by minimizing a loss over the interpolant coefficients as shown in the lower right part of the figure.
• Figure 2: Energy results when simulating Kolmogorov flow with the stochastic interpolants as presented in Section \ref{['subsection:stochastic_interpolants']}.
• Figure 3: Comparison of the optimized interpolant and the interpolant proposed in chen_probabilistic_2024 with $\alpha_\tau = 1-\tau$ and $\beta_\tau = \tau^2$.
• Figure 4: Drift term neural network architecture.
• Figure 5: Kinetic energy results for various generative models.

Theorems & Definitions (2)

• Theorem 3.1: Energy Evolution of the Interpolant
• proof