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A Universal Sampling Framework for Solving Physics-driven Inverse Source Problems

Partial differential equations are central to describing many physical phenomena. Moreover in many applications these phenomena are observed through a sensor network, with the aim of inferring its underlying properties. Here we present a new framework for analysing fields governed by linear partial differential equations. The framework leverages from certain results in sampling and approximation theory to provide a unifying approach for solving a class of inverse source problems. Specifically, we show that the unknown field sources can be recovered from a sequence of so called generalised measurements using multidimensional frequency estimation techniques. Moreover, we show that this sequence of generalised measurements for our physics-driven fields, can be computed by taking linear weighted-sums of the sensor measurements, whereby the exact weights (of the sums) coincide with those that can reproduce multidimensional exponentials from linearly combined translates of a particular prototype function. The prototype function and the desired weights are shown to depend on the Green's function of the underlying field. Based on this new framework we develop practical, noise robust, sensor network strategies for solving the inverse source problem, and then present numerical simulation results to verify their performance.