H.B. Keller Colloquium

Starting with sets of disorganized observations of spatiotemporally evolving systems obtained at different (also disorganized) sets of parameters, I will demonstrate the data-driven derivation of generative, parameter dependent, evolutionary partial differential equation models of the data. We know what observations were made at the same physical location, the same time or the same set of parameter values - knowing neither where the physical location is, nor when the temporal moment is, nor what the parameter values are; this tensor type of data is reminiscent of shuffled (multi)-puzzle tiles . The {\em independent variables} for the evolution equations (their ``space" and ``time") as well as their effective parameters are all ``emergent", i.e. determined in a data-driven way from our disorganized observations of behavior in them.

We use a diffusion map based Questionnaire approach to build a parametrization of our emergent space for the data. This approach iteratively processes the data by successively observing them on the ``space", the ``time" and the ``parameter" axes of a tensor. Once the data are organized, we use ML-based (DNN, GP, Geometric Harmonics) learning to approximate the operators governing the evolution equations in this emergent space. Our illustrative examples are based on (a) a previously developed vertex-plus-signaling model of \textit{Drosophila} embryonic development and (b) networks of heterogeneous coupled oscillators.  This allows us to discuss features of the process like symmetry breaking, translational invariance of the emergent PDE model, and interpretability.