Gaussian processes meet NeuralODEs: a Bayesian framework for learning the dynamics of partially observed systems from...

Area of Science:

• Dynamical systems modeling
• Machine learning for scientific discovery
• Nonlinear system analysis

Background:

• Understanding complex nonlinear dynamical systems is crucial in many scientific fields.
• Traditional modeling approaches often struggle with partial, noisy, and irregular observational data.
• Discovering parsimonious and accurate models from such data remains a significant challenge.

Purpose of the Study:

• To introduce a novel machine learning framework, GP-NODE, for Bayesian model discovery.
• To enable the inference of dynamical system models from limited and imperfect observations.
• To quantify the uncertainty associated with the discovered models and their parameters.

Main Methods:

• Utilizing differentiable programming to integrate ordinary differential equation solvers with gradient propagation.
• Employing Hamiltonian Monte Carlo sampling for Bayesian inference of model parameters.
• Leveraging Gaussian Process priors over system states to exploit temporal correlations.
• Incorporating a sparsity-promoting prior (Finnish Horseshoe) for parsimonious model discovery.

Main Results:

• Demonstrated the effectiveness of GP-NODE across diverse nonlinear dynamical systems, including predator-prey models, systems biology examples, and a high-dimensional human motion system.
• Successfully inferred plausible models with quantified uncertainty from partial, noisy, and irregular data.
• Achieved parsimonious representations of latent dynamics through sparsity promotion.

Conclusions:

• GP-NODE provides a robust and efficient framework for Bayesian model discovery in nonlinear dynamical systems.
• The method effectively handles challenging observational data, offering quantified uncertainty estimates.
• This approach facilitates data-driven prediction and a deeper understanding of complex systems.