This paper introduces a novel framework for physics-aware sparse signal recovery in measurement systems governed by partial differential equations (PDEs). Unlike conventional compressed sensing approaches that treat measurement systems as simple linear systems, our method explicitly incorporates the underlying physics through numerical PDE solvers and automatic differentiation (AD). We present physics-aware iterative shrinkage-thresholding algorithm (PA-ISTA), which combines the computational efficiency of ISTA with accurate physical modeling to achieve improved signal reconstruction. Using optical fiber channels as a concrete example, we demonstrate how the nonlinear Schrödinger equation (NLSE) can be integrated into the recovery process. Our approach leverages deep unfolding techniques for parameter optimization. Numerical experiments show that PA-ISTA significantly outperforms conventional recovery methods. While demonstrated on optical fiber systems, the proposed framework provides a general methodology for physics-aware signal recovery applicable to a wide range of various PDE-governed measurement systems.