When studying observations of chemical reaction dynamics, closed form equations based on a putative mechanism may not be available. Yet when sufficient data from experimental observations can be obtained, even without knowing what exactly the physical meaning of the parameter settings or recorded variables are, data-driven methods can be used to construct minimal (and in a sense, robust) realizations of the system. The approach attempts, in a sense, to circumvent physical understanding, by building intrinsic “information geometries” of the observed data, and thus enabling prediction without physical/chemical knowledge. Here we use such an approach to obtain evolution equations for a data-driven realization of the original system – in effect, allowing prediction based on the informed interrogation of the agnostically organized observation database. We illustrate the approach on observations of (a) the normal form for the cusp singularity, (b) a cusp singularity for the nonisothermal CSTR, and (c) a random invertible transformation of the nonisothermal CSTR, showing that one can predict even when the observables are not “simply explainable” physical quantities. We discuss current limitations and possible extensions of the procedure.

<p>For projects focused on restoration and strengthening of historic structures, information about the origins, magnitude, and stability implications of damage to a structure are required. The objective of this work is to create a novel methodology for understanding the causes of cracks in masonry structures and the resulting effects on global stability. Using Distinct Element Modeling (DEM), the crack patterns of a building can be simulated for a combination of loading scenarios. The results of this method are benchmarked against experimental results and applied to three case studies. The limitations of current physics-based approaches are discussed and a solution using manifold learning is outlined. Manifold learning can be applied to ensembles of crack patterns observed on real or simulated structures to infer damage pathways when the mechanism is unknown. This technique uses a perceptual hashing of the crack patterns to produce an affinity matrix, which is then analyzed by spectral methods to learn a small set of parameters which can describe the ensemble. Because the affinity is derived from a sparse perceptual hash, these descriptors can then be used to interrogate the manifold via a "lifting" operation which reveals the dominant failure modes in the sample.</p>

Starting with sets of disorganized observations of spatially varying and temporally evolving systems, obtained at different (also disorganized) sets of parameters, we demonstrate the data-driven derivation of parameter dependent, evolutionary partial differential equation (PDE) models capable of generating the data. This tensor type of data is reminiscent of shuffled (multi-dimensional) 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 a parametrization of our emergent space/time/parameter 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 machine learning (here, neural networks) to approximate the operators governing the evolution equations in this emergent space. Our illustrative example is based on a previously developed vertex-plus-signaling model of Drosophila embryonic development. This allows us to discuss features of the process like symmetry breaking, translational invariance, and autonomousness of the emergent PDE model, as well as its interpretability. machine learning | generative models | partial differential equations | pattern formation | latent spaces IGK designed research. DWS, FPK, and IGK performed research, analzed data, and wrote the paper. RRC contributed analytic tools and analyzed data.