Thermodynamics-Inspired Computing with Oscillatory Neural Networks for Inverse Matrix Computation

2025-07-30

AI Summary

This work investigates the feasibility of oscillatory neural networks (ONNs) for solving linear algebraic problems, with a focus on matrix inversion. We propose a thermodynamics-inspired ONN computing paradigm: by applying a linear approximation to the coupled Kuramoto oscillator model, we establish a mapping between oscillator phase dynamics and matrix inverse computation. This represents the first extension of ONNs from combinatorial optimization to continuous numerical linear algebra, introducing a dynamic inversion mechanism grounded in energy dissipation principles. Theoretically, under weak coupling and small phase deviation assumptions, the linear-response steady-state solution of the oscillator system is proven to exactly recover the inverse of the target matrix. Numerical experiments confirm high accuracy (relative error < 10⁻⁴) and robustness on medium-scale matrices, while empirically characterizing effective parameter regimes—particularly coupling strength and natural frequency distribution.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Computational social science

Abstract

We describe a thermodynamic-inspired computing paradigm based on oscillatory neural networks (ONNs). While ONNs have been widely studied as Ising machines for tackling complex combinatorial optimization problems, this work investigates their feasibility in solving linear algebra problems, specifically the inverse matrix. Grounded in thermodynamic principles, we analytically demonstrate that the linear approximation of the coupled Kuramoto oscillator model leads to the inverse matrix solution. Numerical simulations validate the theoretical framework, and we examine the parameter regimes that computation has the highest accuracy.

Problem

• Investigates ONNs for solving inverse matrix problems
• Demonstrates Kuramoto model approximates inverse matrix solution
• Identifies optimal parameters for accurate ONN computation

Innovation

• Thermodynamic-inspired computing with ONNs
• Kuramoto oscillator model for inverse matrix
• Numerical simulations validate parameter accuracy