Multimodal oscillator networks learn to solve a classification problem

Abstract

We numerically demonstrate a network of coupled oscillators that can learn to solve a classification task from a set of examples -- performing both training and inference through the nonlinear evolution of the system. We accomplish this by combining three key elements to achieve learning: A long-term memory that stores learned responses, analogous to the synapses in biological brains; a short-term memory that stores the neural activations, similar to the firing patterns of neurons; and an evolution law that updates the synapses in response to novel examples, inspired by synaptic plasticity. Achieving all three elements in wave-based information processors such as metamaterials is a significant challenge. Here, we solve it by leveraging the material multistability to implement long-term memory, and harnessing symmetries and thermal noise to realize the learning rule. Our analysis reveals that the learning mechanism, although inspired by synaptic plasticity, also shares parallelisms with bacterial evolution strategies, where mutation rates increase in the presence of noxious stimuli.

Figures (9)

• Figure 1: Self-learning metamaterial (a) During inference, the metamaterial acts as a mass-spring network. Features are encoded in the excitation amplitude at designated input sites (yellow and purple spheres). If the output amplitude exceeds a threshold, an object is recognized. (b) For the material to learn according to a contrastive learning rule, an identical copy to the lattice in (a) is constructed. The original lattice is operated normally, letting the output site vibrate freely, and is referred to as the free lattice. In the copy, the output is clamped to vibrate at the correct amplitude. At every learning iteration, the coupling springs are updated depending on the difference between the vibrating amplitude of the clamped and free lattices. (c) In the proposed metamaterial, the masses of the free and clamped lattices are mapped to normal modes of each metamaterial site. We refer to these as $\psi_x$ and $\psi_y$ respectively (collectively referred as computational degrees of freedom). The coupling springs are encoded in an additional mode, that is represented by $\sigma$. This mode is parametrically driven at twice its natural frequency (dotted green line) and consequently has two stable phases of oscillation (represented as spin states with blue and red arrows). As a consequence of a nonlinear interaction potential, the effective springs connecting the computational modes depend on the weight state---taking a weak value when neighboring strings are aligned, and a strong value when neighboring weights are oscillating in opposite phases. The learning potential (blue line connecting the modes) causes the weight state to change when clamped and free copies disagree, implementing a contrastive learning rule.
• Figure 2: Single-site learning dynamics. (a) Evolution of the energies in the weight ($\sigma$, middle panel) and computational ($\psi_x$, $\psi_y$, bottom panel) resonators, as a learning pulse is applied (top panel). (b) Probability of flip during a learning protocol as a function of $\psi_X-\psi_Y$ for different temperatures, corresponding to $k_BT$ values of $0.001$ (blue), $0.05$ (orange) and $0.1$ (green). (c) Parametric self-oscillation region (shaded grey area) of the weight degree of freedom $\sigma$; in the shaded region, the weight is bistable and thus has long term memory. The excitation conditions are shown as dashed orange lines and red dot. When $\psi_X-\psi_Y$ is not zero, applying the learning potential shifts the self-oscillation region. The value of $\psi_x-\psi_y$ where $\sigma$ goes out of parametric resonance is shown as a red dashed line in panel b. (d) Probability of spin flip as a function of $\psi_x-\psi_y$, for learning potentials $\mu_m=\mu_{m,0}/n^2$ with $n=1$ (blue), $n=2$ (orange) and $n=3$ (green), illustrating how the step location can be shifted by setting the potential. (e) Probability of spin flip as a function of the protocol durations $\Delta_t=50$ (blue), $\Delta_t=100$ (orange), and $\Delta_t=300$ (green), showing the emergence of probability oscillations at short learning pulses. The dashed curves in panels (c) and (d) show the fit with Eq. \ref{['eqn:switchprobability']}. Throughout the figure, the parameters are $2\omega_l=2$, $\omega_l^2\alpha=0.1$, $\mu_{m,0}=0.005$, and the step duration is $\Delta_T=300$ unless stated otherwise.
• Figure 3: Site-site interactions in a lattice (a) Example metamaterial, consisting of a 4x4 lattice. A harmonic force with frequency $\omega_c$ and amplitude $F_i$ is applied to both clamped and free degrees of freedom at the input site, situated at the top-left corner. The output amplitude $A_o$ is measured at the bottom-right corner. (b) Cumulative density of states (orange) and density of states (blue) as a function of the transmissivity, computed via the effective mass-spring model. The dots correspond to simulations of the transmissivity using the full nonlinear system (Eq. \ref{['eqn:Ising']}) with an excitation force of $10^{-4}$. (c) Weight configurations corresponding to the dots in (b) (d) Evolution of the transmissivity as a function of training iteration for target values $A_T$ of $0$, $1$ and $2$ (dashed line). (e) Output amplitude $A_o$ as a function of the clamping amplitude, after 200 learning iterations, starting from a random configuration. The shaded area represents one standard deviation. The dashed orange line corresponds to an ideal learning response. We observe that, for transmissivity values for which a weight configuration exists, the transmitted amplitude approximately converges to the clamping amplitude. Panels (d) and (e) have been computed by averaging 2000 training runs, with the shaded area representing the standard deviation.
• Figure 4: Iris flower classification using a multifield coherent Ising machine. (a) Flower features contained in the Iris dataset. (b) The features are injected into the multifield coherent Ising machine, by encoding them in the amplitude of harmonic excitations at the computational frequency $\omega_c$. The output is taken at the central site. Positive and negative copies of the signals are applied, as the lattice cannot perform subtractions. The full model consists of three multifield Coherent Ising Machines, corresponding to each of the model classes. The machines learn to produce a high amplitude when excited by a sample of their corresponding class. (c) Evolution of the mean classification accuracy during training. The shaded area corresponds to one standard deviation. (d) Histograms of the classification accuracy computed on an untrained lattice (blue), and after 20 (orange) and 98 (green) training iterations. The training times corresponding to the histograms are indicated as solid dots in panel (c).
• Figure 5: Sensitivity to disorder. (a) Classification accuracy on the Iris dataset as a function of the resonator frequency disorder. (b) Classification accuracy on the Iris dataset as a function of the fraction of defective sites. Both results have been obtained by averaging 16000 simulations. The shaded area corresponds to one standard deviation.