Neural Ising Machines via Unrolling and Zeroth-Order Training

We propose a data-driven heuristic for NP-hard Ising and Max-Cut optimization that learns the update rule of an iterative dynamical system. The method learns a shared, node-wise update rule that maps local interaction fields to spin updates, parameterized by a compact multilayer perceptron with a small number of parameters. Training is performed using a zeroth-order optimizer, since backpropagation through long, recurrent Ising-machine dynamics leads to unstable and poorly informative gradients. We call this approach a neural network parameterized Ising machine (NPIM). Despite its low parameter count, the learned dynamics recover effective algorithmic structure, including momentum-like behavior and time-varying schedules, enabling efficient search in highly non-convex energy landscapes. Across standard Ising and neural combinatorial optimization benchmarks, NPIM achieves competitive solution quality and time-to-solution relative to recent learning-based methods and strong classical Ising-machine heuristics.

💡 Research Summary

The paper introduces Neural Network Parameterized Ising Machines (NPIM), a data‑driven heuristic for solving NP‑hard Ising and Max‑Cut problems. Instead of hand‑crafting update rules for Ising machines, the authors learn a shared, node‑wise update function that maps local interaction fields to spin updates. This function is parameterized by a compact multilayer perceptron (MLP) with only a few hundred parameters. The MLP receives a short history of local fields (length T_c) and, optionally, a time‑dependent set of weights expressed as a linear combination of Fourier basis functions. Consequently, the learned dynamics can exhibit time‑varying schedules and momentum‑like behavior without any explicit design.

Training such recurrent dynamics with back‑propagation is problematic because long roll‑outs cause vanishing/exploding gradients and policy‑gradient methods suffer from severe credit‑assignment noise. To avoid these issues, the authors employ a zeroth‑order evolutionary optimizer (based on Reifenstein et al., 2024). A multivariate Gaussian distribution over the network parameters is maintained (mean θ_x, covariance θ_L). At each epoch, many perturbations θ = θ_x + θ_L v are sampled (v ∼ N(0,I)), each is used to run a stochastic trajectory on a randomly drawn Ising instance, and a reward ρ(traj) is computed. Two reward formulations are explored: (i) final‑energy gap relative to the known optimum, and (ii) best‑energy achieved within a fixed budget. The expected reward R(θ_x,θ_L) is estimated via Monte‑Carlo, and gradient estimates are used to update both mean and covariance. This zeroth‑order scheme sidesteps the need for differentiating through thousands of update steps while still efficiently exploring the parameter space.

Experiments cover standard Ising/Max‑Cut benchmarks (random graphs, planar graphs, G‑set) and neural combinatorial optimization tasks. NPIM is evaluated in two variants: cNPIM (continuous output with tanh) and dNPIM (discrete sign output). Both are compared against classical Ising machines (Coherent Ising Machine, Simulated Bifurcation Machine, Chaotic Amplitude Control), recent learning‑based approaches (GNN‑based Max‑Cut solvers, diffusion samplers), and traditional heuristics (Simulated Annealing, Goemans‑Williamson SDP). Results show that NPIM attains comparable or better average optimality gaps and success rates while using far fewer parameters and comparable wall‑clock time. Notably, the learned schedules automatically start with large, exploratory steps and gradually anneal to finer updates, mirroring hand‑tuned annealing schedules. Momentum‑like terms also emerge, suggesting the network discovers acceleration mechanisms akin to those manually added in physics‑inspired algorithms.

A detailed analysis of the learned parameters reveals that early training emphasizes low‑frequency Fourier modes (large‑scale schedule changes), while higher‑frequency components diminish as training progresses, yielding smoother convergence. The odd‑symmetry constraint (no bias) ensures the update rule respects the Ising problem’s inherent sign symmetry. Visualizations of weight trajectories confirm that the network captures algorithmic structures traditionally designed by experts, offering a degree of interpretability rarely seen in black‑box learned optimizers.

The authors acknowledge limitations: zeroth‑order optimization requires many samples, making training on very large instances computationally expensive; the choice of temporal basis (Fourier vs. Chebyshev vs. Legendre) has modest impact, but the number of modes M is critical; and the current MLP architecture may be insufficient for more complex combinatorial domains. Future work is suggested in meta‑learning for rapid adaptation, richer temporal bases (e.g., wavelets), deeper or graph‑structured networks, and hardware implementations that could directly embed the learned dynamics into photonic or electronic Ising machines.

In summary, the paper demonstrates that a lightweight neural network, trained via evolutionary zeroth‑order methods, can learn effective, interpretable dynamics for Ising machines, achieving performance on par with state‑of‑the‑art physics‑inspired and learning‑based solvers while retaining simplicity, scalability, and a clear algorithmic structure. This work bridges algorithm unrolling, evolutionary optimization, and combinatorial physics, opening new avenues for data‑driven design of combinatorial optimization heuristics.