Chaotic Oscillator Networks for Classification Tasks

Toni Ivas; Georgios Violakis; Roland Richter; Patrik Hoffmann; Sergey Shevchik

nlin.CDMar 6, 2026

Chaotic Oscillator Networks for Classification Tasks

Authors: Toni Ivas, Georgios Violakis, Roland Richter, Patrik Hoffmann, Sergey Shevchik

Organizations: Intelligent Manufacturing Group (IMG) Empa, Thun, Switzerland · Hellenic Mediterranean University Heraklion, Greece

Abstract

Chaotic oscillators have gained significant attention in the research community because of their ability to reproduce and investigate the complex dynamics of real-world phenomena. Recent advances in the design of chaotic oscillator ensembles have led to the development of efficient signal processing frameworks that surpass traditional approaches. However, scaling such systems remains challenging due to the significant increase of computational resources and issues with training convergence. This study advances the state of the art by addressing the problem of data processing with ensembles of nonlinear oscillators that can be scaled up. In our approach, the processing is achieved as an anticipated local resonance or echo in a group of coupled chaotic oscillators, driven by external data input. Local resonance is enabled by tuning the coupling terms between the oscillators, which are approximated using the traditional artificial neural network and adapted to match the input feature distributions. Training the framework entails training this neural network to capture the dynamics of the entire oscillator system. The framework is evaluated using synthetic data and demonstrates an accuracy in machine learning classification task, while patterns recognition and dynamic system identification are also presented here as an extension of the functionality that involves additional modifications. Additionally, the universality of this approach is demonstrated by tests with different connections configurations between the oscillators and their types. The main advantage of the proposed framework is that it avoids hand-crafting explicit coupling terms, which requires expert knowledge and does not scale for large problems. Leveraging standard machine learning components simplifies both training and deployment of oscillator networks, enabling gradient-based optimization.

Figures & tables

$\epsilon$	$0.05$	Time-scale separation (fast/slow)
$a$	$0.5$	Threshold / excitation parameter
$\sigma$	$0.5,\;0.06,\;0.72$	Diffusive coupling / noise strength
$\sigma_{s}$	$0.006,\;0.1$	Spike-rate scaling constant
$T_{\text{span}}$	$(0,\;300)$	ODE integration time window

Table 1: Parameters used in the simulation of the FitzHugh-Nagumo oscillator networks.

Figure 1: Tested network topologies with a) Albert-Barnabasi graph, b) Erdős–Rényi (random) graph, c) Watts-Strogatz (small-world graph) , d) full connectivity used for classification tests.

Figure 2: Network feeding and training schemes: a) The examples from scikit-learn dataset pixel intensity is converted to the pulse-train b) The pulse-train is fed to the network of nonlinear oscillators with specific topology and nonlinear output is trained using the 3-layer neural network. The pattern in (b) corresponds to scikit-learn digits patterns. The coupling in (b) corresponds to $\Phi$ in Eq.(5); c) network training with reservoirs.

Figure 3: The time evolution of the network of FHN oscillators. The network topology corresponds to the one from Figure 1 , d). The following oscillators parameters are used to obtain this dynamics: FitzHugh-Nagumo

a=0.5,\sigma=0.06,e_{0}=0.0,j_{0}=g(t)

. The network includes 1797 nodes and 3227412 edges, here we are displaying only 8 oscillators during time evolution of 2048 steps.

Figure 4: Confusion matrix for classification of scikit-learn dataset using Ridge Regression optimization.

Figure 5: Confusion matrix for classification of the dry bean dataset

Figure 6: The dynamics of the network of oscillators in Watts-Strogatz topology showing learning of XOR gate in 8 node network. Note that output node 5 correctly predicts (

-\pi/2,\pi/2,\pi/2,-\pi/2

Figure 7: Distance function for 5-oscillator Kuramoto network for training over epochs.

Figure 8: a) Test patterns to recognize hand written numbers b) Patterns recognized with fully connected network of 8x8 Kuramoto oscillators recognize the patterns after forcing period $T_{f}>400$ .

Figure 9: The time evolution during pattern recognition by the network of Kuramoto oscillators using Hebbian learning rule Eq.( 20 ) applied to number pattern 1. Note: parameters used $\omega=1.0,\epsilon=100,T_{f}=400,h=0.1$

Panel of Figure 10 — Figure 10: a) Lorenz system with FHN-oscillators network with activation function $swish(x)=x*sigmoid(x)$ with the size of the network with dimension $d=64$ and density of $0.25$ b) The performance for the attractor is very good.

Oscillatory Neural Networks

Oscillatory neural networks use coupled rhythmic units and phase synchronization for learning, sequence processing, memory, and computation.

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May 20, 2026

Winfree Oscillatory Neural Network

Oscillations and synchronization are widely believed to play a fundamental role in representation and computation. However, existing machine learning approaches based on synchronization dynamics have largely been confined to specialized settings such as object discovery, with limited evidence of scalability to standard vision benchmarks or logic reasoning tasks. We propose the Winfree Oscillatory Neural Network (WONN), a dynamical neural architecture based on generalized Winfree dynamics. WONN evolves representations on the torus

(S^1)^d

through structured oscillatory interactions, combining phase-based inductive biases with flexible and hierarchical interaction mechanisms instantiated as either fixed trigonometric mappings or learnable neural networks. We evaluate WONN on image recognition and complex reasoning tasks, including CIFAR, ImageNet, Maze-hard, and Sudoku. Across these domains, WONN achieves competitive or superior performance with strong parameter efficiency. In particular, WONN is, to our knowledge, the first synchronization-based oscillatory architecture to scale competitively to ImageNet-1K. Furthermore, on Maze-hard, WONN achieves 80.1% accuracy using only 1% of the parameters of prior state-of-the-art models. These results suggest that structured oscillatory dynamics provide a scalable and parameter-efficient alternative to conventional neural architectures.

Jiawen Dai, Yue Song

3Shanghai Jiao Tong University · 1Shanghai Qi Zhi Institute · College of AI, Tsinghua University

Dynamic Neural Networks Oscillatory Neural Networks Image Classification

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Mar 10, 2026

Upper Generalization Bounds for Neural Oscillators

Neural oscillators that originate from second-order ordinary differential equations (ODEs) have shown competitive performance in learning mappings between dynamic loads and responses of complex nonlinear structural systems. Despite this empirical success, theoretically quantifying the generalization capacities of their neural network architectures remains undeveloped. In this study, the neural oscillator consisting of a second-order ODE followed by a multilayer perceptron (MLP) is considered. Its upper probably approximately correct (PAC) generalization bound for approximating causal and uniformly continuous operators between continuous temporal function spaces and that for approximating the uniformly asymptotically incrementally stable second-order dynamical systems are derived by leveraging the Rademacher complexity framework. These bounds are further extended to the squared Wasserstein-1 distances between the probability measures of quantities of interest calculated from target causal operators and the corresponding learned neural oscillators. The theoretical results show that the estimation errors grow polynomially with respect to both MLP sizes and the time length, thereby avoiding the curse of parametric complexity. Furthermore, the derived error bounds demonstrate that constraining the Lipschitz constants of the MLPs via loss function regularization can improve the generalization ability of the neural oscillator. Numerical studies considering a Bouc-Wen nonlinear system under stochastic seismic excitation validates the theoretically predicted power laws of the estimation errors with respect to the sample size and time length, and confirms the effectiveness of constraining MLPs' matrix and vector norms in enhancing the performance of the neural oscillator under limited training data.

Zifeng Huang, Konstantin M. Zuev, Yong Xia +1

Institute for Risk and Reliability, Leibniz University Hannover, Callinstraße 34, Hannover, 30167, Germany · Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, California, United States · Joint Research Centre for Marine Infrastructure, Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China +2

Generalization Bounds Neural Network Generalization Oscillatory Neural Networks Statistical Learning Theory

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Jun 22, 2026

The Fractal Neural Operator: Overcoming Spectral Bias in Chaotic Attractors via Prime-Harmonic Weierstrass Encodings

Deep learning models, particularly Transformers and Neural Operators, exhibit a well-documented "spectral bias," effectively acting as low-pass filters that smooth out high-frequency information. While benign in fluid dynamics, this bias is catastrophic for Chaotic Dynamical Systems, where the underlying strange attractor is characterized by fractal geometry and infinite spectral density. We introduce the Fractal Neural Operator (FNO), a novel architecture that utilizes a non-resonant prime number basis to approximate continuous dynamical systems. Unlike geometric encodings (

2^k

), which suffer from spectral gaps and resonance, our Harmonic Weierstrass Encoder injects infinite spectral resolution into the latent space. We demonstrate that FNO extends the valid prediction horizon of the Lorenz-63 system to 347 Lyapunov times, exceeding state-of-the-art Reservoir Computing baselines by a factor of 2.3x. These results suggest that "chaos" is not inherently unpredictable to neural networks, but rather requires non-differentiable, fractal embedding manifolds.

Kanishk Awadhiya

Independent Researcher

Spectral Bias Chaotic Dynamical Systems Neural Operators Chaotic Time-Series Prediction

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Abstract

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