Parameter Estimation in Dynamical Systems

Parameter estimation in dynamical systems centres on the inference of values that specify the evolution of systems governed by mathematical models such as ordinary differential equations. This process addresses the inverse problem, whereby observed data—often sparse, noisy or partially observed—are used to infer the parameters that dictate system behaviour. Approaches range from classical optimisation techniques based on least squares or maximum likelihood to modern Bayesian frameworks that incorporate prior knowledge and quantify uncertainty. Computational methods such as Gaussian process emulation, machine learning surrogates and neural networks have emerged to handle high-dimensional, nonlinear and chaotic models. Identifiability analysis determines which parameters can be uniquely recovered from data, while sensitivity analysis and stability analysis assess how perturbations propagate through the system. Parameter estimation has broad applications including ecological modelling, epidemiology, systems biology, climate prediction and control engineering, offering vital insights into system dynamics and informing decision-making across scientific disciplines.

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Research from all publishers

Recent work in econometrics has introduced adversarial estimation methods, framing parameter inference as a minimax game between a generative model of system output and a discriminator that distinguishes simulated from real data. This approach attains high efficiency and robustness under model misspecification by leveraging neural networks within the discriminator. In computational statistics, new software tools apply manifold-constrained Gaussian processes to noisy and sparse observations of nonlinear ordinary differential equations, enabling joint inference of parameters and unobserved states within a Bayesian framework. Demonstrations include realistic biochemical and epidemiological models, where this method improves estimation accuracy and uncertainty quantification. Advances in the analysis of chaotic dynamics have employed reservoir computing—specifically echo state networks—to infer the Jacobian matrix and compute stability metrics such as Lyapunov exponents and covariant vectors directly from data. These studies open avenues for data-driven stability assessment of complex, high-dimensional systems without explicit knowledge of governing equations.

Parameter Estimation in Dynamical Systems publication trend

The graph below shows the total number of articles in parameter estimation in dynamical systems across all publications each year (not limited to Nature Index journals).

Technical terms

Dynamical system: A mathematical model, often expressed as differential or difference equations, that describes how the state of a system evolves over time.

Parameter estimation: The process of inferring values of unknown model parameters from observational data.

Bayesian inference: A statistical framework that combines prior knowledge and data likelihood to produce a posterior distribution over model parameters.

Gaussian process: A nonparametric prior over functions used to model unknown system trajectories and facilitate parameter inference in a probabilistic manner.

Jacobian: A matrix of first-order partial derivatives of a vector-valued function, used to assess local stability and sensitivity in dynamical systems.

References

1. An Adversarial Approach to Structural Estimation. Econometrica (2023).
2. magi: A Package for Inference of Dynamic Systems from Noisy and Sparse Data via Manifold-Constrained Gaussian Processes. Journal of Statistical Software (2024).
3. Stability analysis of chaotic systems from data. Nonlinear Dynamics (2023).