Problem

Turbulence in fluid dynamics is a complex, high-dimensional chaotic system governed by the Navier-Stokes equations. Understanding its fundamental geometry and structure remains challenging despite advanced numerical methods. A promising approach involves identifying exact solutions, particularly periodic orbits (POs) and relative periodic orbits (RPOs), which are considered the "building blocks" of chaotic dynamics. However, traditional methods for discovering these solutions are computationally intensive and often yield only a narrow subset, failing to fully map the shortest periodic orbits in systems like the 2D Kolmogorov flow at moderate Reynolds numbers, a standard testbed for such studies.

Method

The authors propose a novel approach combining a generative diffusion model with an iterative solver to discover new periodic orbits. They trained a DDIM (Denoising Diffusion Implicit Model), specifically a U-Net architecture, on vorticity time-series data from a direct numerical simulation of turbulent 2D Navier-Stokes equations. Crucially, the training data did not contain periodic trajectories, and the model was only trained on high-dissipation "bursting" events to avoid an abundance of steady-state solutions. The neural network architecture was carefully designed to be equivariant with respect to the known symmetries of the Navier-Stokes equations (rotation, shift-and-reflect, and x-translation), ensuring that generated solutions respect these physical properties. After training, the model's sampling procedure was modified (without retraining weights) to explicitly generate time-periodic or relative-periodic synthetic trajectories by applying custom periodic padding during generation. These synthetic orbits, which served as initial guesses, were then refined into true solutions using a massively parallel, GPU-accelerated Levenberg-Marquardt solver, which iteratively modifies the entire trajectory until it satisfies the governing equations with high precision.

Results

From 2800 synthetic candidate orbits generated by the diffusion model, 597 converged, with 152 of these being unique and non-trivial RPOs. After further refinement and filtering, 111 unique relative periodic orbits (RPOs) with periods less than 3 (T<3) were successfully converged and verified as true solutions to the Navier-Stokes equations. Forty of these had very short periods (T<2), and one even had T<1. These converged solutions demonstrated robustness through reconvergence checks at increased spatial and temporal resolutions. The synthetic guesses, while providing plausible starting points, often differed significantly from the final converged solutions, underscoring the necessity of the iterative solver for achieving exactness. The discovery of these short-period RPOs, many of which were qualitatively different from previously known solutions, reveals a previously unobserved richness in the periodic orbit structure of this system.

Implications

This work demonstrates a significant advance in applying generative artificial intelligence to uncover mathematically meaningful structures in physical models. It establishes generative AI, specifically diffusion models, not as a replacement for traditional numerical simulation or solvers, but as a powerful complementary tool for efficiently navigating and exploring complex solution spaces of nonlinear dynamical systems. The discovered geometrically simple, short-period orbits suggest they could be fundamental "symbols" in a symbolic dynamics for the system, though the sheer number of such orbits (potentially hundreds or thousands) indicates that a complete, rigorously justified periodic orbit expansion for turbulence remains computationally challenging. The methodology highlights the increasing need for interpretability and physical validation in generative AI for scientific applications, combining plausible sampling with strict adherence to governing equations.