Thermodynamic computing with oscillatory neural networks for linear algebra problems

Abstract Physical computing paradigms have recently gained considerable traction. By letting physics take care of the computation, these paradigms offer an energy-efficient alternative to conventional von Neumann architectures. Among these approaches, oscillatory neural networks (ONNs) have emerged as promising candidates, primarily explored as Ising machines for combinatorial optimization and as analog counterparts to Hopfield networks for associative memory. In this work, we investigate a new computational role for ONNs and explore their feasibility in solving linear algebra problems, specifically matrix inversion and linear systems of equations. Inspired by thermodynamic principles, we analytically show that the linear approximation of the coupled Kuramoto oscillator model enables these problems to be mapped onto ONNs. We validate our theoretical framework with numerical simulations and identify parameter regimes for which the ONN yields the highest accuracy. We also provide time-to-solution estimates to assess computational feasibility. These results reveal a previously unexplored application domain for ONNs, going beyond combinatorial optimization and memory tasks and highlighting their potential as a physical solver for linear algebra.