Coupled Dynamical Systems for Solving Linear Inverse Problems

We propose a class of coupled dynamical systems for solving linear inverse problems, treating both the unknown variable and an auxiliary variable representing measurement dynamics as state variables. This framework does not rely on probabilistic modeling or explicit regularization; instead, it achieves noise suppression through deterministic interactions between system variables. We analyze the theoretical properties of the systems, including stability, equilibrium behavior, and convergence for the linear system, and equilibrium stability for the two nonlinear variants. The nonlinear extensions incorporate state-dependent mechanisms that preserve equilibrium stability while enhancing convergence and robustness in practice. Numerical experiments illustrate the effectiveness of the proposed approach in estimating the unknown variable and mitigating measurement noise.