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-fidelity 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.