Solving Sudoku using oscillatory neural networks

TL;DR

This work develops a physics-inspired approach to solving Sudoku by encoding the puzzle into a graph of coupled oscillators and using Kuramoto-phase dynamics to drive synchronization toward valid digit assignments. It introduces a phase-mapping scheme, a tailored weight-mapping strategy for known/unknown cells and Sudoku constraints, and metrics (order parameter and time-to-settle) to evaluate performance. Compared with a baseline Hopfield network, the ONN demonstrates higher solution rates and better scaling across multiple grid sizes, albeit with reduced performance as puzzle size and unknown density grow. The results suggest that exploiting the natural dynamics of oscillatory systems can be a hardware-efficient alternative for constraint satisfaction and related combinatorial optimization problems.

Abstract

We explore the capabilities of physical computing with Oscillatory Neural Networks (ONN) to solve combinatorial optimization problems. To solve Sudokus with ONNs, we define a novel mapping strategy that utilizes the unique characteristics of the computation paradigm. The problem is encoded through a puzzle specific graph-embedding, which implements the constraints through different subgraphs. These subgraphs are then combined into a single adjacency matrix, which allows the natural dynamics of the phases of coupled oscillators to find a solution to the puzzle. We model the phase dynamics of the ONN by means of the Kuramoto differential equation. This novel approach is then compared to the well-established iterative method to solve Sudoku already used in binary Hopfield networks (HNN). Solving optimization problems typically requires a large amount of energy to solve on conventional hardware. Therefore, we are motivated to explore the mapping of Sudoku from a theoretical point of view to establish the validity of this approach. The simulation results show that the novel ONN mapping outperforms the established HNN methodology.

Figures (10)

• Figure 1: Example of an Oscillatory Neural Network, made up of five elements. Each node represents the dynamical state of the phase of an oscillator.
• Figure 2: An example of an unsolved Sudoku puzzle. The empty cells have to filled in by the player according to the puzzle constraints. The filled cells have a fixed value.
• Figure 3: An example of a solved $4 \times 4$ Sudoku grid. The graph representations of Figs. \ref{['fig:subgrid']}, \ref{['fig:column']}, \ref{['fig:row']} are based on the $4 \times 4$ example.
• Figure 4: Digits of a $4\times4$ Sudoku mapped onto the unit circle with their respective phase values.
• Figure 5: Example graph representation of the first Sudoku subgrid, dubbed the subgrid and notated symbolically as $\mathcal{S}$.