Linking Gaussian process regression with data-driven manifold embeddings for nonlinear data fusion

In statistical modelling with Gaussian process regression, it has been shown that combining (few) high-fidelity data with (many) low-fidelity data can enhance prediction accuracy, compared to prediction based on the few high-fidelity data only. Such information fusion techniques for multi-fidelity data commonly approach the high-fidelity model f(h)(t) as a function of two variables (t, s), and then use f(l)(t) as the s data. More generally, the high-Wfidelity model can be written as a function of several variables (t, s(1), s(2)....); the low-fidelity model f(l) and, say, some of its derivatives can then be substituted for these variables. In this paper, we will explore mathematical algorithms for multi-fidelity information fusion that use such an approach towards improving the representation of the high-fidelity function with only a few training data points. Given that f(h) may not be a simple function-and sometimes not even a function-of f(l), we demonstrate that using additional functions of t, such as derivatives or shifts of f(l), can drastically improve the approximation of f(h) through Gaussian processes. We also point out a connection with 'embedology' techniques from topology and dynamical systems. Our illustrative examples range from instructive caricatures to computational biology models, such as Hodgkin-Huxley neural oscillations.