M2P 2023

Lipschitz Stabilised Autoencoders to Study the Intrinsic Dimensionality of Dynamical Systems and Build Data-driven Models.

  • Liu, Haibo (Inria Paris)
  • Lombardi, Damiano (Inria Paris)
  • Boulakia, Muriel (Universit√© de Versailles)

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The present work deals with data-driven modelling. The context which we are interested in is the following. A set of partial noisy observations of a dynamical system is available (typically sampled in time components of a system state). These data are supposed to be generated by a dynamical system. We make the hypothesis that, besides the noise, the difference between the trajectories is due to the variation of a certain (unknown) number of hidden parameters. The first goal is to understand how many parameters are responsible for the observed variability. The idea consists in using an auto-encoder to perform such a task. However, a classical auto-encoder might suffer from a methodological issue. Having fixed a number of hidden units, we can make the autoencoder complexity grow and have a satisfactory fitting of the trajectories. The price to pay is that the maps built by the auto-encoder have a very large Lipschitz constant. This phenomenon (which we will explain theoretically based on some recent results in approximation theory [1]) implies that when we will use the mappings, we will have a potentially unstable behaviour. By introducing a penalisation of the Lipschitz constants of the auto-encoder map, we will be able to retrieve the number of hidden units making it possible to retrieve the trajectories in a stable way. This provides a way to uncover the intrinsic dimensionality of the set of observation curves. The second goal consists in exploiting this in order to build a data-driven model and eventually understand the relationship between this description and a physical based description of the same phenomenon, based on a different parametrisation, chosen a priori on the basis of first principles. Some numerical results on non-linear models are presented. REFERENCES [1] Cohen, A., Devore, R., Petrova, G., & Wojtaszczyk, P. (2022). Optimal stable nonlinear approximation. Foundations of Computational Mathematics, 22(3), 607-648.