Programmable k-local Ising Machines and all-optical Kolmogorov-Arnold Networks on Photonic Platforms

Photonic computing promises energy-efficient acceleration for optimization and learning, yet discrete combinatorial search and continuous function approximation have largely required distinct devices and control stacks. Here we unify k-local Ising optimization and optical Kolmogorov-Arnold network (KAN) learning on a single photonic platform, establishing a critical convergence point in optical computing. We introduce an SLM-centric primitive that realizes, in one stroke, all-optical k-local Ising interactions and fully optical KAN layers. The key idea is to convert the structural nonlinearity of a nominally linear scatterer into a per-window computational resource by adding a single relay pass through the same spatial light modulator: a folded 4f relay re-images the first Fourier plane onto the SLM so that each selected clique or channel occupies a disjoint window with its own second pass phase patch. Propagation remains linear in the optical field, yet the measured intensity in each window becomes a freely programmable polynomial of the clique sum or projection amplitude. This yields native, per clique k-local couplings without nonlinear media and, in parallel, the many independent univariate nonlinearities required by KAN layers, all trainable with in-situ physical gradients using two frames (forward and adjoint). We outline implementations on spatial photonic Ising machines, injection-locked vertical cavity surface emitting laser (VCSEL) arrays, and Microsoft analog optical computers; in all cases the hardware change is one extra lens and a fold (or an on-chip 4f loop), enabling a minimal overhead, massively parallel route to high-order Ising optimization and trainable, all-optical KAN processing on one platform.