(132c) Learning What to Learn: Some Data-Driven Twists in Linking System Identification, Manifold Learning, and (possibly) Causality Considerations

We explore the use of manifold learning techniques (and in particular, Diffusion Maps and their Alternating Diffusion extension) to detect common underlying latent variables across observations of dynamical systems by different observers/measuring instruments. This data-driven latent space then allows us to deploy several different machine learning techniques in order to build nonlinear observers across quantities of interest, and even learn evolution laws for these quantities from temporal or spatiotemporal data. It even allows for the possibility of testing and establishing causal relations (in the ``correlational” rather than the ``interventional” sense of causality) across observables.

Along the same lines, manifold learning is implemented in order to merge
multi-fidelity data, i.e. observations of the same process from models
or measuring devices of varying accuracy. Specifically, the goal is to
implement data-driven methodologies, in this case Alternating Diffusion
maps [1] and autoencoders[2], in order to develop and apply “nonlinear
filtering” of the low-fidelity observations in an effort to leverage the
availability of heterogeneous data.

We also explore the use of manifold learning techniques for the systematic discovery of useful embedding spaces for disorganized heterogeneous observations: finding the ``right space” -the right independent variables- in which we can learn the effective dynamics as a partial differential equation using machine learning techniques in the form black or gray box models. Some analogies between these tools and the adaptive control dynamics studied by Erik Ydstie will be discussed.

[1] Ronen Talmon, Hau-Tieng Wu, Latent common manifold learning with
alternating diffusion: analysis and applications, arXiv:1602.00078v2
[2] Erez Peterfreund, Ofir Lindenbaum, Felix Dietrich, Tom Bertalan,
Matan Gavish, Ioannis G. Kevrekidis, Ronald R. Coifman, LOCA: LOcal
Conformal Autoencoder for standardized data coordinates,
arXiv:2004.07234