Is it possible to do functional gaussian process regression using fitrgp?

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My understanding is that typical gaussian process regression assumes that direct observations are made: e.g. what is the altitude of the surface of the earth for a given longitude and latitude? However, some problems involve indirect observations, as in remote sensing, resistor networks, etc, in which case an "effective" property is measured. Is there a straightforward way to implement functional gaussian process regression (i.e. indirect observations that depend on multiple datapoints) in fitrgp? If so, I imagine it would involve specifying a custom covariance matrix, perhaps a custom basis function, and maybe even specifying fmincon as an optimizer with a custom gradient (i.e. functional gradient), but the implementation is elusive to me. The specific equations I'm interested in are equilibrium equations. In other words, if the problem were formulated as a linear problem, A*x = b, with A === sparse design matrix (encodes the indirect observations), then y is a vector of zeros.
I'm also open to other non-parametric techniques, preferably ones that can give some measure of uncertainty in the predicted values.
In the case of using a different technique (e.g. regress, fitrlinear), I've considered using a design matrix, but this matrix will be "wide", with more unknowns than equations and requires some regularization.

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