Data-driven guessing and gluing of unstable periodic orbits

TY - JOUR

T1 - Data-driven guessing and gluing of unstable periodic orbits

AU - Beck, Pierre

AU - Parker, Jeremy P.

AU - Schneider, Tobias M.

N1 - ©2025 American Physical Society

PY - 2025/8/4

Y1 - 2025/8/4

N2 - Unstable periodic orbits (UPOs) are believed to be the underlying dynamical structures of spatiotemporal chaos and turbulence. Finding these UPOs is, however, notoriously difficult. Matrix-free loop convergence algorithms deform entire space-time fields (loops) until they satisfy the evolution equations. Initial guesses for these robust variational convergence algorithms are thus periodic space-time fields in a high-dimensional state space, rendering their generation highly challenging. Usually guesses are generated with recurrency methods, which are most suited to shorter and more stable periodic orbits. Here we propose an alternative, data-driven method for generating initial guesses, enabled by the periodic nature of the guesses for loop convergence algorithms: While the dimension of the space used to discretize fluid flows is prohibitively large to construct suitable initial guesses, the dissipative dynamics will collapse onto a chaotic attractor of far lower dimension. We use an autoencoder to obtain a low-dimensional representation of the discretized physical space for the one-dimensional Kuramoto-Sivashinksy equation in chaotic and hyperchaotic regimes. In this low-dimensional latent space, we construct loops based on the latent POD modes with random periodic coefficients, which are then decoded to physical space and used as initial guesses. These loops are found to be realistic initial guesses and, together with variational convergence algorithms, these guesses help us to quickly converge to UPOs. We further attempt to “glue” known UPOs in the latent space to create guesses for longer ones. This gluing procedure is successful and points towards a hierarchy of UPOs where longer UPOs shadow sequences of shorter ones.

AB - Unstable periodic orbits (UPOs) are believed to be the underlying dynamical structures of spatiotemporal chaos and turbulence. Finding these UPOs is, however, notoriously difficult. Matrix-free loop convergence algorithms deform entire space-time fields (loops) until they satisfy the evolution equations. Initial guesses for these robust variational convergence algorithms are thus periodic space-time fields in a high-dimensional state space, rendering their generation highly challenging. Usually guesses are generated with recurrency methods, which are most suited to shorter and more stable periodic orbits. Here we propose an alternative, data-driven method for generating initial guesses, enabled by the periodic nature of the guesses for loop convergence algorithms: While the dimension of the space used to discretize fluid flows is prohibitively large to construct suitable initial guesses, the dissipative dynamics will collapse onto a chaotic attractor of far lower dimension. We use an autoencoder to obtain a low-dimensional representation of the discretized physical space for the one-dimensional Kuramoto-Sivashinksy equation in chaotic and hyperchaotic regimes. In this low-dimensional latent space, we construct loops based on the latent POD modes with random periodic coefficients, which are then decoded to physical space and used as initial guesses. These loops are found to be realistic initial guesses and, together with variational convergence algorithms, these guesses help us to quickly converge to UPOs. We further attempt to “glue” known UPOs in the latent space to create guesses for longer ones. This gluing procedure is successful and points towards a hierarchy of UPOs where longer UPOs shadow sequences of shorter ones.

KW - Low-dimensional models

KW - Pattern formation

KW - Spatiotemporal chaos

KW - Turbulence

KW - Coherent structures

KW - High dimensional systems

KW - Chaos & nonlinear dynamics

KW - Machine learning

U2 - 10.1103/9vjc-g86s

DO - 10.1103/9vjc-g86s

M3 - Article

C2 - 40954763

SN - 1063-651X

VL - 112

SP - 24203

JO - Physical Review E

JF - Physical Review E

IS - 2-1

M1 - 024203

ER -