Koopman Autoencoders with Continuous-Time Latent Dynamics for Fluid Dynamics Forecasting

TL;DR

The paper tackles long-horizon turbulent-flow forecasting by introducing a continuous-time Koopman autoencoder that learns latent linear dynamics governed by $\frac{dz}{dt} = \mathbf{K}_{\text{cont}}(\phi) z$, enabling variable-time-step rollouts. It combines a dual-stream Transformer encoder, a physics-conditioned latent Koopman operator, and a CNN decoder, and trains with a rollout-based objective plus physics-informed regularization. The continuous-time formulation achieves accuracy competitive with diffusion baselines while delivering orders-of-magnitude faster inference, and it exhibits zero-shot temporal generalization due to the matrix-exponential evolution $z_{\tau} = \exp(\mathbf{K}\tau) z_0$. The method demonstrates robustness across incompressible and transonic CFD benchmarks, offering practical impact for real-time or large-scale flow simulations and design optimization, with limitations related to spectral bias and the assumption of global linear latent dynamics.

Abstract

Data-driven surrogate models have emerged as powerful tools for accelerating the simulation of turbulent flows. However, classical approaches which perform autoregressive rollouts often trade off between strong short-term accuracy and long-horizon stability. Koopman autoencoders, inspired by Koopman operator theory, provide a physics-based alternative by mapping nonlinear dynamics into a latent space where linear evolution is conducted. In practice, most existing formulations operate in a discrete-time setting, limiting temporal flexibility. In this work, we introduce a continuous-time Koopman framework that models latent evolution through numerical integration schemes. By allowing variable timesteps at inference, the method demonstrates robustness to temporal resolution and generalizes beyond training regimes. In addition, the learned dynamics closely adhere to the analytical matrix exponential solution, enabling efficient long-horizon forecasting. We evaluate the approach on classical CFD benchmarks and report accuracy, stability, and extrapolation properties.

Figures (8)

• Figure 1: Architecture overview. The history encoder and present encoder (top) feed into the latent state $z_{t_0}$, followed by a dynamics rollout and decoding at each step.
• Figure 2: Qualitative validation rollouts for extrapolation regimes. (Left) Vorticity prediction for the incompressible wake flow in the high-Reynolds-number regime ($Inc_{high}, Re=1000$). (Right) Pressure prediction for the transonic cylinder flow in the low-Mach extrapolation regime ($Tra_{ext}, Ma=0.50$).
• Figure 3: Comparison between numerical RK4 integration and the analytical matrix exponential solution of the learned continuous-time latent dynamics. Results are shown for (Left) incompressible flow vorticity at $Re=1000$ and (Right) transonic flow pressure at $Ma=0.50$.
• Figure 4: Direct comparison of inference results using the matrix exponentiation and integrator at different time steps. All columns display the flow's evolution at the same physical time and each row represents a different evolution scheme: first one is the direct matrix exponentiation, while the subsequent ones use the RK4 integrator with different time steps.
• Figure 5: Field-averaged mean squared error (MSE) comparison between KAE and ACDM for incompressible flows. Error bars denote one standard deviation across test trajectories. Our method consistently achieves lower error and reduced variance, with the performance gap widening in the high-Re regime where chaotic dynamics dominate.