arXiv subjects

Carlos E. Pérez De Jesús

Publications and source records attributed to Carlos E. Pérez De Jesús.

4 recordsLinked to original sources

arXiv2025

Building symmetries into data-driven manifold dynamics models for complex flows: application to two-dimensional Kolmogorov flow

Data-driven reduced-order models of the dynamics of complex flows are important for tasks related to design, understanding, prediction, and control. Many flows obey symmetries, and the present work illustrates how these can be exploited to yield highly efficient low-dimensional data-driven models for chaotic flows. In particular, incorporating symmetries both guarantees that the reduced order model automatically respects them and dramatically increases the effective density of data sampling. Given data for the long-time dynamics of a system, and knowing the set of continuous and discrete symmetries it obeys, the first step in the methodology is to identify a "fundamental chart", a region in the state space of the flow to which all other regions can be mapped by a symmetry operation, and a set of criteria indicating what mapping takes each point in state space into that chart. We then find a low-dimensional coordinate representation of the data in the fundamental chart with the use of an autoencoder architecture that also provides an estimate of the dimension of the invariant manifold where data lie. Finally, we learn dynamics on this manifold with the use of neural ordinary differential equations. We apply this method, denoted "symmetry charting" to simulation data from two-dimensional Kolmogorov flow in a chaotic bursting regime. This system has a continuous translation symmetry, and discrete rotation and shift-reflect symmetries. With this framework we observe that less data is needed to learn accurate data-driven models, more robust estimates of the manifold dimension are obtained, equivariance of the NSE is satisfied, better short-time tracking with respect to the true data is observed, and long-time statistics are correctly captured.

cs.LG↗

arXiv2024

Data-driven Koopman operator predictions of turbulent dynamics in models of shear flows

The Koopman operator enables the analysis of nonlinear dynamical systems through a linear perspective by describing time evolution in the infinite-dimensional space of observables. Here this formalism is applied to shear flows, specifically the minimal flow unit plane Couette flow and a nine-dimensional model of turbulent shear flow between infinite parallel free-slip walls under a sinusoidal body force (Moehlis et al., New J. Phys. 6, 56 (2004)). We identify a finite set of observables and approximate the Koopman operator using neural networks, following the method developed by Constante-Amores et al., Chaos, 34(4), 043119 (2024). Then the time evolution is determined with a method here denoted as "Projected Koopman Dynamics". Using a high-dimensional approximate Koopman operator directly for the plane Couette system is computationally infeasible due to the high state space dimension of direct numerical simulations (DNS). However, the long-term dynamics of the dissipative Navier-Stokes equations are expected to live on a lower-dimensional invariant manifold. Motivated by this fact, we perform dimension reduction on the DNS data with a variant of the IRMAE-WD (implicit rank-minimizing autoencoder-weight decay) autoencoder (Zeng et al., MLST 5(2), 025053 (2024)) before applying the Koopman approach. We compare the performance of data-driven state space (neural ordinary differential equation) and Projected Koopman Dynamics approaches in manifold coordinates for plane Couette flow. Projected Koopman Dynamics effectively captures the chaotic behavior of Couette flow and the nine-dimensional model, excelling in both short-term tracking and long-term statistical analysis, outperforming other frameworks applied to these shear flows.

physics.flu-dyn↗

arXiv2023

Autoencoders for discovering manifold dimension and coordinates in data from complex dynamical systems

While many phenomena in physics and engineering are formally high-dimensional, their long-time dynamics often live on a lower-dimensional manifold. The present work introduces an autoencoder framework that combines implicit regularization with internal linear layers and $L_2$ regularization (weight decay) to automatically estimate the underlying dimensionality of a data set, produce an orthogonal manifold coordinate system, and provide the mapping functions between the ambient space and manifold space, allowing for out-of-sample projections. We validate our framework's ability to estimate the manifold dimension for a series of datasets from dynamical systems of varying complexities and compare to other state-of-the-art estimators. We analyze the training dynamics of the network to glean insight into the mechanism of low-rank learning and find that collectively each of the implicit regularizing layers compound the low-rank representation and even self-correct during training. Analysis of gradient descent dynamics for this architecture in the linear case reveals the role of the internal linear layers in leading to faster decay of a "collective weight variable" incorporating all layers, and the role of weight decay in breaking degeneracies and thus driving convergence along directions in which no decay would occur in its absence. We show that this framework can be naturally extended for applications of state-space modeling and forecasting by generating a data-driven dynamic model of a spatiotemporally chaotic partial differential equation using only the manifold coordinates. Finally, we demonstrate that our framework is robust to hyperparameter choices.

cs.LG↗

arXiv2023

Data-driven low-dimensional dynamic model of Kolmogorov flow

Reduced order models (ROMs) that capture flow dynamics are of interest for decreasing computational costs for simulation as well as for model-based control approaches. This work presents a data-driven framework for minimal-dimensional models that effectively capture the dynamics and properties of the flow. We apply this to Kolmogorov flow in a regime consisting of chaotic and intermittent behavior, which is common in many flows processes and is challenging to model. The trajectory of the flow travels near relative periodic orbits (RPOs), interspersed with sporadic bursting events corresponding to excursions between the regions containing the RPOs. The first step in development of the models is use of an undercomplete autoencoder to map from the full state data down to a latent space of dramatically lower dimension. Then models of the discrete-time evolution of the dynamics in the latent space are developed. By analyzing the model performance as a function of latent space dimension we can estimate the minimum number of dimensions required to capture the system dynamics. To further reduce the dimension of the dynamical model, we factor out a phase variable in the direction of translational invariance for the flow, leading to separate evolution equations for the pattern and phase. At a model dimension of five for the pattern dynamics, as opposed to the full state dimension of 1024 (i.e. a 32x32 grid), accurate predictions are found for individual trajectories out to about two Lyapunov times, as well as for long-time statistics. Further small improvements in the results occur at a dimension of nine. The nearly heteroclinic connections between the different RPOs, including the quiescent and bursting time scales, are well captured. We also capture key features of the phase dynamics. Finally, we use the low-dimensional representation to predict future bursting events, finding good success.

cs.LG↗