Physics-Aware Sparse Signal Recovery Through PDE-Governed Measurement Systems

TL;DR

Sparse signal recovery in PDE governed measurement systems is addressed by PA-ISTA, which differentiates through a NLSE based forward map computed with SSFM and optimized via deep unfolding. The method formulates a Lasso like objective $F(\bm{s}) = \|\bm{y} - \hat{f}(\bm{s})\|_2^2 + \lambda \|\bm{s}\|_1$ and updates via complex ISTA with a physics aware gradient. Key contributions include the nested AD challenge and the store-and-replay solution for DU, plus demonstrated gains in MSE and SER on optical fiber channels. The framework offers a general methodology for physics informed sparse recovery across PDE governed systems with potential broad impact in sensing and communications.

Abstract

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.

Figures (8)

• Figure 1: Scope for physics-aware sparse signal recovery.
• Figure 2: Visualization of signal propagation through an optical fiber. The left panel displays the intensity profile of the input signal $|U(t,0)|^2$ at the fiber input. The center panel shows a three-dimensional visualization of the signal intensity $|U(t,z)|^2$, illustrating how the waveform evolves as it propagates along the fiber length $z$. The right panel presents the measured intensity profile $|U(t,L)|^2$ at the fiber output, including the effects of measurement noise.
• Figure 3: Signal recovery process of PA-ISTA.
• Figure 4: Nested structure of gradient computation (A) and store-and-replay method (B).
• Figure 5: Example of signal recovery. PA-ISTA with DU-optimized parameters were used (Blue:real part, Red: imaginary part). The SNR was set to 15 dB.