Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approach

We address a three-tier computational approach for the construction of coarse-grained bifurcation diagrams from spatio-temporal data produced by microscopic simulators using machine learning. In the first step, we exploit manifold learning and in particular parsimonious Diffusion Maps to identify the intrinsic dimension of the manifolds where the emergent dynamics evolve and feature selection for the parametrization of these manifolds. In the second step, based on the selected features we learn the right-hand-side of the effective partial differential equations (PDEs) using two machine learning schemes, namely Feed-forward Neural Networks (FNNs) and Random Projection Networks (RPNNs). Finally, based on the learned black-box PDE model, we construct the corresponding bifurcation diagram, thus exploiting numerical bifurcation theory algorithms. For our illustrations, we implemented the proposed method to construct the one-parameter bifurcation diagram of the 1D FitzHugh-Nagumo PDEs from data generated by Lattice-Boltzman (LBM) numerical simulations.