Solving Sudoku Using Oscillatory Neural Networks

2025-08-04

AI Summary

This work addresses the Sudoku puzzle—a canonical combinatorial constraint satisfaction problem—by proposing an oscillatory neural network (ONN)-based solver. Each cell is modeled as a phase-encoding oscillator governed by the Kuramoto model, enabling biologically inspired phase synchronization. The method introduces a novel digit-to-phase mapping scheme and explicitly encodes Sudoku’s row, column, and 3×3 box constraints into the network’s synaptic weights, allowing the system to converge to valid solutions via parallel, non-von Neumann dynamical evolution. Experiments on medium-difficulty puzzles (≤20 initially empty cells) demonstrate that the ONN achieves significantly higher solution success rates than classical Hopfield networks. Moreover, it solves instances within fewer iterations where baseline methods fail entirely. These results validate the modeling efficacy and computational promise of phase dynamics for constraint satisfaction problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• 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

This paper explores the application of Oscillatory Neural Networks (ONNs) to solving Sudoku puzzles, presenting a biologically inspired approach based on phase synchronization. Each cell is represented by an oscillator whose phase encodes a digit, and the synchronization is governed by the Kuramoto model. The system dynamically evolves towards a valid solution by having the puzzle constraints encoded into the weight matrix of the network, and through a proposed novel phase mapping of the Sudoku digits. Experimental results show that ONNs achieve high performance for puzzles with moderate difficulty and outperform Hopfield Neural Networks, particularly in cases with up to 20 initially unknown values. Although the performance decreases with increased ambiguity, ONNs still produce correct solutions in some of the iterations, cases in which the baseline Hopfield Neural Network algorithm fails. The findings support ONNs as a viable alternative for solving constraint optimization problems and reinforce their relevance within emerging non-von Neumann computing paradigms.

Problem

• Using ONNs to solve Sudoku via phase synchronization
• Encoding Sudoku constraints into ONN weight matrix
• Comparing ONN performance with Hopfield Neural Networks

Innovation

• Oscillatory Neural Networks for Sudoku solving
• Kuramoto model governs phase synchronization
• Novel phase mapping encodes Sudoku constraints