From synthetic turbulence to true solutions: A deep diffusion model for discovering periodic orbits in the Navier-Stokes equations

TL;DR

This paper introduces a deep diffusion model trained on turbulence data to discover new periodic solutions of the Navier-Stokes equations, demonstrating AI's potential to explore complex physical systems.

Contribution

The work presents a novel application of generative diffusion models to uncover previously unknown periodic orbits in turbulent fluid dynamics, combining AI with traditional solvers.

Findings

Generated 111 new periodic orbits in turbulence data.

Synthetic trajectories can be refined into true solutions.

Revealed complex structure of solutions in Navier-Stokes system.

Abstract

Generative artificial intelligence has shown remarkable success in synthesizing data that mimic complex real-world systems, but its potential role in the discovery of mathematically meaningful structures in physical models remains underexplored. In this work, we demonstrate how a generative diffusion model can be used to uncover previously unknown solutions of a nonlinear partial differential equation: the two-dimensional Navier-Stokes equations in a turbulent regime. Trained on data from a direct numerical simulation of turbulence, the model learns to generate time series that resemble physically plausible trajectories. By carefully modifying the temporal structure of the model and enforcing the symmetries of the governing equations, we produce synthetic trajectories that are periodic in time, despite the fact that the training data did not contain periodic trajectories. These synthetic trajectories are then refined into true solutions using an iterative solver, yielding 111 new periodic orbits (POs) with very short periods. Our results reveal a previously unobserved richness in the PO structure of this system and suggest a broader role for generative AI: not as replacements for simulation and existing solvers, but as a complementary tool for navigating the complex solution spaces of nonlinear dynamical systems.