Deep learning to discover and predict dynamics on an inertial manifold

Linot, Alec J.; Graham, Michael D.

doi:10.1103/physreve.101.062209

Cited by 67 publications

(75 citation statements)

References 28 publications

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“…That is, mappings χ and χ exist such that h = χ(u) and u = χ(h). In the machine learning literature, χ and χ correspond to the the encoder and decoder, respectively, of a so-called undercomplete autoencoder structure [25], as we further discuss below. It should be noted that there is no guarantee that M can be globally represented with a cartesian representation in d M dimensions.…”

Section: Introductionmentioning

confidence: 99%

Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations

Linot,

Graham

2021

Preprint

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Dissipative partial differential equations that exhibit chaotic dynamics tend to evolve to attractors that exist on finite-dimensional manifolds. We present a data-driven reduced order modeling method that capitalizes on this fact by finding the coordinates of this manifold and finding an ordinary differential equation (ODE) describing the dynamics in this coordinate system. The manifold coordinates are discovered using an undercomplete autoencoder -a neural network (NN) that reduces then expands dimension. Then the ODE, in these coordinates, is approximated by a NN using the neural ODE framework. Both of these methods only require snapshots of data to learn a model, and the data can be widely and/or unevenly spaced. We apply this framework to the Kuramoto-Sivashinsky for different domain sizes that exhibit chaotic dynamics. With this system, we find that dimension reduction improves performance relative to predictions in the ambient space, where artifacts arise. Then, with the low-dimensional model, we vary the training data spacing and find excellent short-and long-time statistical recreation of the true dynamics for widely spaced data (spacing of ∼0.7 Lyapunov times). We end by comparing performance with various degrees of dimension reduction, and find a "sweet spot" in terms of performance vs. dimension.

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Section: Introductionmentioning

confidence: 99%

Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations

Linot,

Graham

2021

Preprint

Self Cite

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“…In this sense, the proposed procedure is not only able to determine an upper bound to the quantity of variables that are actually needed to describe the original trajectory. The idea of "compressing" the signal and only keeping relevant information is not new in machine learning (and specific machine learning approach to attractor size estimation can be found in literature 31 ) but, unlike some automatic dimensionalreduction approaches (e.g. autoencoder neural networks, see Appendix B), the technique employed in this paper also provides some direct hint on which variables may be expected to be relevant in modelling the macroscopic dynamics.…”

Section: Discussionmentioning

confidence: 99%

“…While even pure model-free methods can be very efficient [24][25][26] , other approaches aim to blend physical information with data-only techniques [27][28][29] . An important role is played by those machine learning methods attempting to extract an effective lowdimensional dynamics, for instance by employing autoencoder based networks 30,31 .…”

Section: Introductionmentioning

confidence: 99%

“…Remarkably, such an approach to model reconstruction also yields relevant physical information about the underlying physical process, implying that the quality of forecasts and the physical insight are strongly interconnected. We therefore explore the possibility to overcome the separation between a purely result-oriented approach and theoretical investigation, which is a promising direction for machine learning based approaches 31,[35][36][37] .…”

Section: Introductionmentioning

confidence: 99%

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Using machine-learning modeling to understand macroscopic dynamics in a system of coupled maps

Borra

Baldovin

2021

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Machine-learning techniques not only offer efficient tools for modeling dynamical systems from data but can also be employed as frontline investigative instruments for the underlying physics. Nontrivial information about the original dynamics, which would otherwise require sophisticated ad hoc techniques, can be obtained by a careful usage of such methods. To illustrate this point, we consider as a case study the macroscopic motion emerging from a system of globally coupled maps. We build a coarse-grained Markov process for the macroscopic dynamics both with a machine-learning approach and with a direct numerical computation of the transition probability of the coarse-grained process, and we compare the outcomes of the two analyses. Our purpose is twofold: on the one hand, we want to test the ability of the stochastic machine-learning approach to describe nontrivial evolution laws as the one considered in our study. On the other hand, we aim to gain some insight into the physics of the macroscopic dynamics. By modulating the information available to the network, we are able to infer important information about the effective dimension of the attractor, the persistence of memory effects, and the multiscale structure of the dynamics.

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“…In combination with nlPCA, this method has successfully been applied by e.g. Linot & Graham [13], who also included temporal advancement in the reduced space. The method is only valid when the amplitude of the first Fourier mode is non-zero, since the shift of a sample with zero amplitude would become infinite.…”

Section: Introductionmentioning

confidence: 99%

Symmetry-Aware Autoencoders: s-PCA and s-nlPCA

Kneer,

Sayadi,

Sipp

et al. 2021

Preprint

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Nonlinear principal component analysis (nlPCA) via autoencoders has attracted attention in the dynamical systems community due to its larger compression rate when compared to linear principal component analysis (PCA). These model reduction methods experience an increase in the dimensionality of the latent space when applied to datasets that exhibit globally invariant samples due to the presence of symmetries. In this study, we introduce a novel machine learning embedding in the autoencoder, which uses spatial transformer networks and Siamese networks to account for continuous and discrete symmetries, respectively. The spatial transformer network discovers the optimal shift for the continuous translation or rotation so that invariant samples are aligned in the periodic directions. Similarly, the Siamese networks collapse samples that are invariant under discrete shifts and reflections. Thus, the proposed symmetry-aware autoencoder is invariant to predetermined input transformations dictating the dynamics of the underlying physical system. This embedding can be employed with both linear and nonlinear reduction methods, which we term symmetryaware PCA (s-PCA) and symmetry-aware nlPCA (s-nlPCA). We apply the proposed framework to 3 fluid flow problems: Burgers' equation, the simulation of the flow through a step diffuser and the Kolmogorov flow to showcase the capabilities for cases exhibiting only continuous symmetries, only discrete symmetries or a combination of both.

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Deep learning to discover and predict dynamics on an inertial manifold

Cited by 67 publications

References 28 publications

Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations

Data-Driven Reduced-Order Modeling of Spatiotemporal Chaos with Neural Ordinary Differential Equations

Using machine-learning modeling to understand macroscopic dynamics in a system of coupled maps

Symmetry-Aware Autoencoders: s-PCA and s-nlPCA

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