Variational mean-field algorithm for efficient inference in large systems of stochastic differential equations.

Area of Science:

• Dynamical Systems and Control Theory
• Computational Statistics
• Machine Learning

Background:

• Dynamical systems modeled by ordinary stochastic differential equations (SDEs) present significant inference challenges.
• Approximate inference in high-dimensional SDEs is computationally intensive.
• Existing methods often struggle with scalability and complexity.

Purpose of the Study:

• To introduce a computationally efficient Gaussian variational mean-field approximation for SDEs.
• To reduce the complexity of approximate inference in nonlinear dynamical systems.
• To enable scalable state and parameter estimation for high-dimensional problems.

Main Methods:

• Developed a Gaussian variational mean-field approximation for SDEs.
• Expressed variational free energy as a functional of marginal moments of a Gaussian process.
• Restricted moment equations to piecewise polynomial functions for reduced complexity.
• Applied the algorithm to state and parameter estimation in nonlinear systems.

Main Results:

• The proposed method significantly reduces the complexity of approximate inference for SDE models.
• The computational complexity is made comparable to that of discrete-time hidden Markov models.
• Demonstrated effectiveness on nonlinear problems with up to 1000-dimensional state vectors.
• Empirically compared results with established inference methodologies.

Conclusions:

• The Gaussian variational mean-field approximation offers an efficient and scalable approach for inference in SDEs.
• This method provides a practical solution for state and parameter estimation in complex, high-dimensional dynamical systems.
• The approach bridges the gap between continuous-time SDE inference and established discrete-time models.