Physics-Aware Linearized ADMM and Its Unrolling

2026-06-01

AI Summary

This work addresses the high computational cost of partial differential equation (PDE)-based measurement models in inverse problems of signal processing by proposing a physics-aware linearized alternating direction method of multipliers (PA-LADMM). The algorithm linearizes PDE-constrained subproblems, requiring only a single evaluation of the PDE solver and its adjoint gradient per iteration, and for the first time integrates PDE-based measurement models into a linearized ADMM framework. By further incorporating deep unfolding techniques, the method leverages supervised data to learn algorithmic parameters, enhancing reconstruction performance while preserving theoretical convergence guarantees. Experiments on compressed sensing in fiber-optic communications and anisotropic diffusion-based image restoration demonstrate that the proposed approach achieves significant improvements in both accuracy and computational efficiency.

Abstract

Recently, partial differential equations (PDEs) have been used to directly model the measurement process in signal processing, although their evaluation is costly. In this paper, we propose a novel alternating direction method of multipliers (ADMM)-based algorithm called physics-aware linearized ADMM (PA-LADMM) for inverse problems from PDE-based measurement processes. The key idea is the linearization of the subproblem with PDEs, leading to a cost-efficient update rule that calls only a PDE solver and its gradient evaluation per iteration. The algorithm has a theoretical convergence guarantee under certain conditions. In addition, we combine it with deep unfolding to unroll the PA-LADMM and train its internal parameters using supervised data. Two distinct experiments, compressed sensing with optical fiber communication and image restoration from noisy anisotropic diffusion, demonstrated the effectiveness of the proposed algorithms.

Problem

• inverse problems
• PDE-based measurement
• computational cost
• signal processing
• physics-aware modeling

Innovation

• physics-aware
• linearized ADMM
• PDE-based inverse problems
• deep unfolding
• computational efficiency