(284h) Emergent Evolution Equations from (multi-)Puzzle Tiles, with a Drosophila Embryonic Development Example

Starting with sets of disorganized observations of spatiotemporally evolving systems obtained at different (also disorganized) sets of input conditions, we 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 input conditions - knowing neither where the physical location is, nor when the temporal moment is, nor what the input conditions are; we call this tensor type of data "multi-puzzle tiles".

The 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 the parametrization of our emergent space. This approach iteratively processes the data by successively observing them on the "space", the "time" and the "parameter" axes of a tensor [1-3]. This is followed by neural-network-based learning of the operators governing the evolution equations in this emergent space [4]. Our illustrative example is based on a previously developed vertex-plus-signaling model of Drosophila embryonic development [5]. This allows us to discuss features of the process like symmetry breaking, translational invariance of the PDE model, and interpretability.

References
[1] J.I. Ankenman, Geometry and Analysis of Dual Networks on Questionnaires, Ph.D. thesis, Yale University (2014).
[2] O. Yair, R. Talmon, R.R. Coifman, and I.G. Kevrekidis, Proc. Natl. Acad. Sci. U. S. A. 114, E7865 (2017).
[3] D.W. Sroczynski, O. Yair, R. Talmon, and I.G. Kevrekidis, Isr. J. Chem. 58, 787 (2018).
[4] F.P. Kemeth, T. Bertalan, T. Thiem, F. Dietrich, S.J. Moon, C.R. Laing, and I.G. Kevrekidis, Learning emergent PDEs in a learned emergent space, ArXiv (2020).
[5] M. Misra, B. Audoly, I.G. Kevrekidis, and S.Y. Shvartsman, Biophys. J. 110, 1670 (2016).