Coarse-grained and emergent distributed parameter systems from data

For example, For many systems of interest in physics or engineering, in the case of collective particle motion, a natural choice we are given a fine-scale description of the system evolution, for such an independent variable would be the coordinates e.g. at the particle-based or agent-based level; yet the system of the space in which the particles move, and the coarsegrained exhibits large-scale, coarse-grained, spatiotemporal patterns PDE would involve the spatial derivatives of some which may well be captured by a set of unknown effective, unknown, coarse dependent variables. We assume that these coarse-grained possibly emergent PDEs. Such reduced, effective unknown dependent variables capture the local collective PDEs, when they exist and can be derived (whether (possibly averaged) statistical features of the particles, and mathematically, or in a data-driven fashion) can serve as hence can be written in terms of the local particle distribution cheap surrogate models, drastically facilitating computationintensive observations. We use manifold learning to extract tasks like prediction, optimization, uncertainty these coarse nonlinear observables from mining local particle quantification and even control.