Identifying recurrent flows in high-dimensional dissipative chaos from low-dimensional embeddings

TL;DR

This work tackles the difficulty of locating non‑chaotic invariant solutions (UPOs) in high‑dimensional dissipative chaos by embedding the attractor into a low‑dimensional latent space via an autoencoder trained with physics and tangent losses. The latent dynamics are derived from the physical equations using the chain rule and automatic differentiation, enabling a loop‑convergence optimization that finds latent UPOs without time‑marching the chaotic system. The authors demonstrate a one‑to‑one correspondence between latent UPOs and physical UPOs for both the Kuramoto–Sivashinsky equation and Kolmogorov flow (2D NSE), with small period discrepancies and preserved invariant sets and statistics. By decoupling attractor dimension from discretization and reducing the search space, this approach permits efficient discovery of long UPOs and suggests a path toward first‑principles turbulence descriptions via ergodic theory.

Abstract

Unstable periodic orbits (UPOs) are the non-chaotic, dynamical building blocks of spatio-temporal chaos, motivating a first-principles based theory for turbulence ever since the discovery of deterministic chaos. Despite their key role in the ergodic theory approach to fluid turbulence, identifying UPOs is challenging for two reasons: chaotic dynamics and the high-dimensionality of the spatial discretization. We address both issues at once by proposing a loop convergence algorithm for UPOs directly within a low-dimensional embedding of the chaotic attractor. The convergence algorithm circumvents time-integration, hence avoiding instabilities from exponential error amplification, and operates on a latent dynamics obtained by pulling back the physical equations using automatic differentiation through the learned embedding function. The interpretable latent dynamics is accurate in a statistical sense, and, crucially, the embedding preserves the internal structure of the attractor, which we demonstrate through an equivalence between the latent and physical UPOs of both a model PDE and the 2D Navier-Stokes equations. This allows us to exploit the collapse of high-dimensional dissipative systems onto a lower dimensional manifold, and identify UPOs in the low-dimensional embedding.

Figures (6)

• Figure 1: Schematic representation of the high-dimensional dynamics of a driven dissipative PDE collapsing on a chaotic attractor $\mathcal{A}$ that can be embedded in a much lower-dimensional manifold $\mathcal{M}$ (left). Through data-driven dimensionality reduction (autoencoders in our case), points from the high-dimensional physical space $\boldsymbol{u}\in\mathbb{R}^N$ are mapped to points $\boldsymbol{h}\in\mathbb{R}^{N_h}$ in the low-dimensional latent space ($N_h \ll N$). The flow vectors (blue) in the latent space are defined through chain rule, by pulling the physical flow into the latent space. Based on this flow, we implement a loop convergence algorithm, where an initial guess loop (grey dashed) is deformed into an unstable periodic orbit (red solid in the latent space). The flow vectors are always tangent to the periodic solution, which is not the case for the guess.
• Figure 2: Reconstruction performance of the autoencoder, and the latent dynamics$\partial_t\boldsymbol{h}$. A typical trajectory of the KSE, simulated on a $N_x = 64$ grid, is plotted in the top left plot of panel A with the temporal derivative in the top right. The middle row shows the autoencoder output of each individual snapshot of the ground truth (left), and the temporal derivative of each output (right). Relative errors are on average $\sim\mathcal{O}(10^{-5})$ and $\sim\mathcal{O}( 10^{-3})$ respectively, and are given in log-scale in the bottom row. Panel B shows a projection on the first two latent coordinates $(h_1, h_2)$ of the latent attractor (grey) and latent flow vectors $\partial_t\boldsymbol{h}$ (black) evaluated on an encoded physical periodic orbit (blue) with period $T\approx77.4$. We observe that the flow vectors are always tangent to the periodic orbit. Integrating an arbitrary initial condition on the attractor with $\partial_t\boldsymbol{h}$ yields trajectories like the blue one in panel C. The trajectory stays on the attractor (grey background) and covers it in a stable manner. In panel D, we compare time-integration with $\partial_t\boldsymbol{u}$ (top) and $\partial_t\boldsymbol{h}$ (middle) from one same initial condition. The temporal evolution of the relative difference in the bottom (blue) matches the error growth $\propto\exp(t/t_L)$ given by the Lyapunov time $t_L \approx 21.6$ (black), thus showing no signature of inaccuracy beyond the divergence between nearby trajectories intrinsic to the chaotic system. This indicates that the latent dynamics accurately captures the original system.
• Figure 3: Equivalence between latent UPOs and physical UPOs in the KSE. Panel A shows an example of a decoded latent UPO (top) with period $T_{lat} \approx 190.84$ and the physical UPO it converges to (bottom) with period $T_{phys} \approx 190.79$. A latent projection of the UPO pair is shown in panel B, with the latent chaotic attractor in grey, the latent UPO in red and the encoded physical UPO in blue dashed. There are no visible differences in both the physical and latent representations. The periods of all unique 195 UPO pairs agree very well, as shown by the $y = x$ trend in panel C, with an average relative difference between $T_{phys}$ and $T_{lat}$ of $\approx2.4\times10^{-4}$. In 16 cases, the latent UPO does not converge to a physical UPO, but to a very low local minimum of the cost function. Slight perturbation of the control parameter $L$ results in a UPO as shown in panel D. Through pseudo-arclength continuation we obtain the bifurcation diagram of panel E, which shows the saddle-node bifurcation (blue) and the resulting ghost state (the local minimum of the cost function --- red dashed) passing through the original parameter $L=39$ (black dashed).
• Figure 4: Reconstruction performance of the autoencoder for Kolmogorov flow. Panel A shows test snapshots (top) and the autoencoder output (bottom), ordered in increasing dissipation $D$ (left to right). The relative differences $\epsilon(\boldsymbol{\omega}, \check{\boldsymbol{\omega}})$ are on average 1%, and 0.16%, 0.14%, 0.71% and 4.4% for the individual snapshots. The temporal derivative for the same snapshots and the autoencoder outputs are shown in panel B. Visual differences appear only for the high dissipation snapshot. The relative differences $\epsilon(\partial_t\boldsymbol{\omega}, \partial_t\check{\boldsymbol{\omega}})$ are 8% on average, and 2.98%, 4.10%, 7.04% and 20.8% for the individual snapshots. Histograms cleary2025 of the normalized dissipation against $\epsilon(\boldsymbol{\omega}, \check{\boldsymbol{\omega}})$ and $\epsilon(\partial_t\boldsymbol{\omega}, \partial_t\check{\boldsymbol{\omega}})$ are displayed in panels C and D.
• Figure 5: Latent dynamics$\partial_t\boldsymbol{h}$of the autoencoder. Encoded physical snapshots (grey) and a trajectory integrated with $\partial_t\boldsymbol{h}$ (blue) in the latent space coordinates $(h_1, h_2)$ are depicted in panel A. The latent attractor shows an octogonal shape in this projection, relating to the discrete symmetries of the system PBK2021. The latent trajectory does not leave the attractor and starts covering it, indicating ergodicity. A similar conclusion can be drawn from the physical projection onto normalized dissipation $D$ and energy input $E$, shown in panel B. In particular, the dissipations of physical snapshots and decoded latent trajectories, $D$ and $D_h$ respectively, agree in distribution, indicated by the $y=x$ trend of the quantile-quantile plot of panel C. Snapshots of trajectories integrated with $\partial_t\boldsymbol{\omega}$ and $\partial_t\boldsymbol{h}$ from one same initial condition for $t \in\{15,30,45,60\}$ are shown in panel D, with the evolution of the relative error $\epsilon(\boldsymbol{\omega},\ \check{\boldsymbol{\omega}})$ shown in panel E. The red points mark the errors for the plotted snapshots.