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We address a three-tier data-driven approach to solve the inverse problem incomplex systems modelling from spatio-temporal data produced by microscopicsimulators using machine learning. In the first step, we exploit manifoldlearning and in particular parsimonious Diffusion Maps using leave-one-outcross-validation (LOOCV) to both identify the intrinsic dimension of themanifold where the emergent dynamics evolve and for feature selection over theparametric space. In the second step, based on the selected features, we learnthe right-hand-side of the effective partial differential equations (PDEs)using two machine learning schemes, namely shallow Feedforward Neural Networks(FNNs) with two hidden layers and single-layer Random ProjectionNetworks(RPNNs) which basis functions are constructed using an appropriaterandom sampling approach. Finally, based on the learned black-box PDE model, weconstruct the corresponding bifurcation diagram, thus exploiting the numericalbifurcation analysis toolkit. For our illustrations, we implemented theproposed method to construct the one-parameter bifurcation diagram of the 1DFitzHugh-Nagumo PDEs from data generated by $D1Q3$ Lattice Boltzmannsimulations. The proposed method was quite effective in terms of numericalaccuracy regarding the construction of the coarse-scale bifurcation diagram.Furthermore, the proposed RPNN scheme was $\sim$ 20 to 30 times less costlyregarding the training phase than the traditional shallow FNNs, thus arising asa promising alternative to deep learning for solving the inverse problem forhigh-dimensional PDEs.

journal of scientific computing

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