Identifiability and amortized inference limitations in Kuramoto models

Emma Hannula; Jana de Wiljes; Matthew T. Moores; Heikki Haario; Lassi Roininen

stat.APMar 23, 2026

Identifiability and amortized inference limitations in Kuramoto models

Authors: Emma Hannula, Jana de Wiljes, Matthew T. Moores, Heikki Haario, Lassi Roininen

Organizations: LUT University, Yliopistonkatu 34, Lappeenranta, 53850, Finland · TU Ilmenau, Ehrenbergstraße 29, Ilmenau, 98693, Germany · University of Cambridge, Herschel Road, Cambridge, CB3 9AL, United Kingdom

Abstract

Bayesian inference is a powerful tool for parameter estimation and uncertainty quantification in dynamical systems. However, for nonlinear oscillator networks such as Kuramoto models, widely used to study synchronization phenomena in physics, biology, and engineering, inference is often computationally prohibitive due to high-dimensional state spaces and intractable likelihood functions. We present an amortized Bayesian inference approach that learns a neural approximation of the posterior from simulated phase dynamics, enabling fast, scalable inference without repeated sampling or optimization. Applied to synthetic Kuramoto networks, the method shows promising results in approximating posterior distributions and capturing uncertainty, with computational savings compared to traditional Bayesian techniques. These findings suggest that amortized inference is a practical and flexible framework for uncertainty-aware analysis of oscillator networks.

Figures & tables

Figure 1: Illustration of training and inference phases in amortized Bayesian inference.

Figure 2: Three simulations from the Kuramoto system with $N=100$ oscillators with given initial values and frequencies, randomized by $\zeta\sim\mathcal{N}(0,10^{-2})$ . The system is simulated for the duration of 1000 time steps.

Panel of Figure 3 — Figure 3: Example of a three-node closed network connectivity graph and its adjacency matrix.

Variables
$r$	$\Psi$
$\text{mean}(\sin(\psi))$	$\text{std}(\sin(\psi))$
$\text{mean}(\cos(\psi))$	$\text{std}(\cos(\psi))$

Table 1: Handcrafted summary statistics.

	Simple network	Complex network
Oscillators ( $N$ )	100	3
Estimated parameters ( $\kappa$ )	1	6
Training samples ( $n_{t}$ )	$2^{12}$	$2^{17}$
Validation samples ( $n_{v}$ )	$2^{6}$	$2^{7}$
Epochs	50	150
Batches	64	128

Table 2: Fixed training parameters.

Figure 4: Estimated vs true

\kappa

values with

n_{d}

samples from

n_{v}

simulations.

Figure 5: Posterior distribution for

\kappa

with

n_{d}

samples.

Panel of Figure 6 — Figure 6: Goodness of fit evaluation.

	Simple network	Complex network
Metric	$\kappa$	$\kappa_{1}$	$\kappa_{2}$	$\kappa_{3}$	$\kappa_{4}$	$\kappa_{5}$	$\kappa_{6}$
NRMSE	0.0584	0.3139	0.3136	0.2336	0.2206	0.2687	0.2889
Posterior Contraction	0.9874	0.4469	0.2975	0.6673	0.8335	0.5725	0.5779
Calibration Error	0.0683	0.0463	0.0181	0.0089	0.0178	0.0423	0.0651

Table 3: Evaluation of model performances for simple and complex network using normalized root mean square error, posterior contraction and calibration error.

Figure 7: Posterior estimates for the complex network with true parameters $\kappa_{1}=0.6$ , $\kappa_{2}=0.5$ , $\kappa_{3}=0.4$ , $\kappa_{4}=0.3$ , $\kappa_{5}=0.2$ and $\kappa_{6}=0.1$ . The true parameter values are marked with red dashed lines or circular red dots.

Kuramoto Model

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Jan 12, 2026J/K move · Enter open · S save

Neural Architectures for Amortized Bayesian Inference: Statistical Foundations and Empirical Assessments

Roy Shivam Ram Shreshtth, Arnab Hazra, Gourab Mukherjee

Department of Statistics and Data Science, Indian Institute of Technology Kanpur, Kanpur, India 208016 · Marshall School of Business, University of Southern California, Los Angeles, United States 90089-0809

Uncertainty Quantification Amortized Inference Bayesian Inference

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Since the turn of the century, approximate Bayesian inference has steadily evolved as new computational techniques have been incorporated to handle increasingly complex, large-scale predictive problems. The recent success of deep neural networks and foundation models has now given rise to a new paradigm in statistical modeling, in which Bayesian inference can be amortized through large-scale learned predictors. In amortized inference, substantial computation is required at the beginning to train a neural network, but it can subsequently produce approximate posteriors or predictions at much lower computational cost across a wide range of tasks. While the typical Bayesian inference procedures are computationally expensive due to repeated likelihood calculations and Monte Carlo steps for each new dataset, amortized inference provides a much lower computational cost at deployment. Despite the growing popularity of amortized inference, its statistical interpretation and position within Bayesian inference remain poorly explored. In this paper, we present a statistical perspective on several major neural architectures, including feedforward networks, Deep Sets, and Transformers, and examine how they naturally support amortized Bayesian inference. We explore how these models perform structured approximation and also probabilistic reasoning in ways that yield controlled generalization error throughout a wide range of deployment scenarios, and how these properties can be harnessed for Bayesian computation. Via simulation studies, we evaluate the accuracy, robustness, and uncertainty quantification of amortized inference across varying sample sizes, varying noise distributional families, varying sparsity levels, and multimodality, highlighting its strengths and limitations.

Abstract

Kuramoto Model

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