Inferring solutions of dierential equations using noisy multi-delity data

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Neural-net-induced Gaussian process regression for function approximation and PDE solution (2018, methodology): “…mean and covariance function k u ( x , x (cid:48) ) , and the source term is another GP that depends on u , whose covariance function is written as [6] Denoting by X u ( N u by d in matrix) the collection of the boundary evaluation points { x u,j } and by X f ( N f by d in matrix) the collection…”
Machine Learning of Space-Fractional Differential Equations (2018, methodology): “Building on an earlier work [26] where Gaussian process regression was used to infer solutions of equations, a Gaussian process framework was developed in [27] for parametric learning of linear diﬀerential equations, of the form (1.1) L p 1 ,...,p k u = f given small data on f and u .”
A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems (2019, methodology): “Several methods have been developed to estimate such correlations, such as the response surface models [11, 12], polynomial chaos expansion [13, 14], Gaussian process regression (GPR) [6, 8, 9, 15], artiﬁcial neural networks [16], and moving least squares [17, 18].”