Nanoparticle Networks for Neuromorphic Computing

Physical computing leverages complex dynamical systems for energy-efficient data processing. In this work, we present a neuromorphic architecture based on metallic nanoparticles interconnected by molecular junctions on a $\text{SiO}_2$/Si substrate. We demonstrate that surrounding static control electrodes transform this nanoparticle network from a passive reservoir into a tunable nonlinear dynamical system. By analyzing how these electrodes route simple one-dimensional voltage inputs into multidimensional signal responses, we establish three core design rules to maximize computational performance. First, operating near the system's cutoff frequency achieves an optimal balance between nonlinear charge tunneling and linear capacitive memory. Second, tuning the underlying $\text{SiO}_2$ thickness sets the electrostatic screening length and dictates the memory type. Thick oxide layers reduce the screening length, causing networks larger than this length to transition into a persistent, non-volatile-like regime. Conversely, networks smaller than the screening length exhibit only fading memory. Third, introducing structural disorder via heterogeneous molecular junctions overcomes inherent limits on expressivity. While a network's computational expressivity scales with its physical size, it is ultimately capped by the screening length. Breaking internal spatial symmetries with localized disorder bypasses this saturation, allowing control voltages to independently manipulate specific signal amplitudes and phases, universally maximizing performance for dynamic neuromorphic applications.

Attention by Synchronization in Coupled Oscillator Networks

We address transformer attention on energy-constrained physical substrates. Softmax attention requires exponentiation and global reduction, operations with high energy cost on von Neumann hardware and no natural physical analog. We show that Kuramoto synchronization dynamics (which arise in electrical, mechanical, superconducting, and charge-density-wave oscillator arrays, among other physical systems) implement a well-defined attention operation without either. The resulting mechanism, fixed-query oscillator attention, replaces softmax's arithmetic with the equilibration of a gradient flow on the sphere: queries are learned anchors fixed on the sphere, and free oscillators evolve under Kuramoto-Lohe dynamics until they settle at positions encoding attention weights via cosine similarity. Because the computation is equilibration, it requires no exponentiation; the only global operation is an affine normalization at readout. The fixed point is provably unique and globally attractive from almost every initial condition, a guarantee that holds across every physical realization. Empirically, at the minimal hardware configuration (oscillator dimension $d_{\mathrm{osc}}$ = 2), oscillator attention outperforms softmax on keyword spotting (+1.00 pp) and on subject-verb agreement (+5.27 pp on hard sentences, with zero training failures versus one in five for softmax). On causal language modeling, where softmax retains an advantage, oscillator attention closes the gap as $d_{\mathrm{osc}}$ grows: from +11.09 PPL at $d_{\mathrm{osc}}$ = 2 to +2.98 PPL at $d_{\mathrm{osc}}$ = 32 on WikiText-2, and from +2.39 PPL at $d_{\mathrm{osc}}$ = 2 to +0.57 PPL at $d_{\mathrm{osc}}$ = 32 on TinyStories. The main objective of this work is not to replace softmax in software but to provide a mathematically grounded blueprint for accurate attention on physical substrates.