Simulating Stochastic Population Dynamics: The Linear Noise Approximation Can Capture Nonlinear Phenomena

Revisit a SIR System: Using the Dynamical System Approach to Detect Slow-Fast Oscillations

Data-Driven Identification of Stochastic Dynamical Systems

Identifying stochastic dynamical systems from observational data remains a major challenge in applied mathematics and engineering, particularly when complex systems are influenced by random perturbations and incomplete empirical information. This comprehensive review aims to examine state-of-the-art data-driven methods for discovering governing equations, estimating parameters, and predicting the behavior of stochastic dynamical systems. The review systematically analyzes key methodological approaches, including Sparse Identification of Nonlinear Dynamics (SINDy), Dynamic Mode Decomposition (DMD) and its extensions, Koopman operator theory, neural ordinary differential equations, and Bayesian inference. Each approach is evaluated in terms of its theoretical foundations, computational requirements, robustness to noise, and applicability to different classes of stochastic systems. Drawing on numerical experiments and real-world case studies, the findings show that no single method consistently outperforms others across all scenarios. Instead, hybrid approaches that integrate physics-informed constraints with machine learning demonstrate the strongest potential for advancing data-driven system identification. The review concludes that future research should address real-time identification, uncertainty quantification, and the integration of multi-fidelity data sources to improve the reliability and scalability of stochastic system modeling. This work contributes a comprehensive framework for guiding researchers and practitioners in selecting and implementing appropriate identification methods for stochastic dynamical systems.

Nonlinear Exchange Dynamics for Independent Sets

In recent years, nonlinear dynamics derived from kinetic theory have gained attention in the context of sampling configurations of spin systems such as the Ising model. We focus on nonlinear dynamics for the hard-core model, a canonical spin system with hard constraints that specifies a distribution over independent sets in a graph, weighted by their sizes. We explore two distinct types of nonlinear dynamics: the mean-field dynamics, which preserves the density (or average size) of independent sets, and the single-site dynamics, which preserves the marginal vector (i.e., the occupancy probabilities of the vertices). These dynamics are natural stochastic processes for sampling from the hard-core model with a specified density or marginal vector, respectively, both of which are canonical instances of maximum entropy distributions that have been studied in various contexts. In contrast to linear Markov chains, there is a significant lack of a fundamental theoretical framework for nonlinear dynamics. We develop foundational theoretical tools for analyzing nonlinear dynamics within the context of the hard-core model. We establish almost linear convergence of both the mean-field and single-site dynamics at sufficiently low density through novel coupling arguments. We also establish exponential decay of relative entropy for the mean-field dynamics all the way up to the critical density. Additionally, we design new algorithms for sampling from the hard-core distribution with either a specified density or a specified marginal vector. These algorithms are based on a related linear Markov chain, called the particle-system dynamics and inspired by the so-called Kac's program, that approximates the associated nonlinear dynamics. As we demonstrate in the paper, they are comparable in time complexity, but simpler to implement, than traditional approaches based on learning parameter values.

Approximation of Nonlinear Dynamical Systems via Parametric Rational Interpolation Methods

Parameter identification for predator-prey system with sparse data

Parameter identification from observations of dynamical systems is a fundamental problem in population biology. Mechanistic models of ecological systems rely on optimization methods that require accurate initial guesses to guarantee convergence. In ecological applications, datasets contain observation noise and are collected at sparse time points. This sparsity creates irregular likelihoods that cause standard optimization methods to struggle, while the ordinary differential equation solvers can become stiff or unstable in certain regions of the parameter space. These instabilities cause long running times or runtime errors. Here we present a computational framework for parameter identification that addresses these numerical instabilities by employing Natural Gradient Ascent, and we apply it to the classical Lotka-Volterra predator-prey model. We exploit the non-dimensionalization of the ordinary differential equations to treat scaling factors as nuisance parameters, reducing the dimensionality of the optimization problem. To prevent the solver step from becoming small, we implement an adaptive solver that switches between two independent second-order equations derived from the two components of the model. This approach allows Natural Gradient Ascent to converge in fewer iterations and with more stability than standard gradient ascent or BFGS methods. This framework provides a reliable method for parameter estimation in ecology when data is limited. The method can be generalized to other dynamical systems as long as the different components of the system do not become numerically problematic at the same time.

Uncertainty Quantification in Stochastic Dynamical Gene Regulatory Networks

The dynamics of gene regulatory networks are governed by intrinsic noise, stemming from the random nature of biochemical reactions, and by extrinsic noise, arising from fluctuations in cellular components and environmental conditions. Together, these sources can compromise the reliability of predictive computational models if not properly accounted for, and capturing both effects within a single framework remains a non-trivial task in computational biology. In this work, we propose an uncertainty quantification framework that addresses these two contributions jointly: intrinsic stochasticity is described through a partial integro-differential equation (PIDE) for the protein probability density function, whereas extrinsic noise is represented as parametric uncertainty in the kinetic parameters. The propagation of the uncertainty is carried out via an intrusive polynomial chaos expansion (PCE), in which the PCE coefficients are obtained from a stochastic Galerkin projection of the PIDE, yielding a coupled deterministic system that is solved with standard numerical methods. We illustrate the approach on a positive autoregulatory gene network with one and two uncertain kinetic parameters. The proposed approach accurately reproduces the mean, variance, and full protein probability density function, including the bimodal distributions, at a substantially lower computational cost.