Mixing Artificial and Natural Intelligence: From Statistical Mechanics to AI and Back to Turbulence

TL;DR

This work surveys how Artificial Intelligence intersects with turbulence and broader sciences through a framework grounded in non-equilibrium statistical mechanics. It highlights diffusion-model paradigms, physics-informed AI, and Lagrangian closures as concrete avenues where AI can learn, interpret, and extend turbulence physics, including neural tetrad closures and SPH-based neural Lagrangian LES. The authors discuss theoretical and practical implications—from MCMC-inspired sampling and time-reversal to U-turn and bridge-diffusion techniques—that enable more efficient data generation, extrapolation, and reduced-order modeling. They propose a future in which AI-guided reduced models, multi-fidelity Lagrangian approaches, and diffusion-based dynamics accelerate discovery and practical control of turbulent flows, while also prompting new hypotheses in turbulence through AI-driven exploration. Overall, the piece argues for deep integration of AI with statistical hydrodynamics to push both AI methodology and turbulence science forward, with concrete directions in diffusion-augmented modeling, Lagrangian closures, and physics-guided multi-fidelity frameworks.

Abstract

The paper reflects on the future role of AI in scientific research, with a special focus on turbulence studies, and examines the evolution of AI, particularly through Diffusion Models rooted in non-equilibrium statistical mechanics. It underscores the significant impact of AI on advancing reduced, Lagrangian models of turbulence through innovative use of deep neural networks. Additionally, the paper reviews various other AI applications in turbulence research and outlines potential challenges and opportunities in the concurrent advancement of AI and statistical hydrodynamics. This discussion sets the stage for a future where AI and turbulence research are intricately intertwined, leading to more profound insights and advancements in both fields.

Figures (9)

• Figure 1: GPT-4 prompt: Illustration of the University of Arizona's bobcat mascot swimming butterfly style in a turbulent river. (The entire prompt was used to generate the image without providing additional information such as color, symmetry, or viewing direction.)
• Figure 2: Illustration, from behjoo_u-turn_2023, of the basic Score Based Diffusion (left) and U-turn (right) concepts. The U-turn involves an earlier transition from the forward to the reverse process, optimizing sample generation.
• Figure 3: This visualization showcases generative capability of the space-time bridge-diffusion model from behjoo_space-time_2024 to synthesize high-fidelity images. The diffusion model was trained on the MNIST dataset, an established benchmark in the machine learning domain for handwritten digit recognition, which consists of $70,000$ images with dimensions of $28\times 28$ pixels.
• Figure 4: Schematic illustration of Lagrangian trajectories of a pair of particles relevant for computations of passive scalar correlations in Eq. (\ref{['eq:theta-2']}).
• Figure 5: Schematic illustration of the minimal representation of a blob via a tetrad (four black dots/particles) or equivalently the (orange) ellipsoid. Geometry of the tetrad/ellipsoid, described in terms of the ellipsoid's tensor of inertia, ${\bm g}$, is evolving in time (not shown). (The blue dot marks the tetrad center of inertia.)