Data-driven prediction of multistable systems from sparse measurements.

Area of Science:

• Complex Systems
• Computational Mathematics
• Applied Physics

Background:

• Multistable systems exhibit multiple stable states, making their long-term behavior prediction challenging.
• Sparse spatial measurements are often the only feasible data acquisition method for complex systems.
• Predicting asymptotic states is vital for understanding pattern formation and biological dynamics.

Purpose of the Study:

• To develop a data-driven, semi-supervised classification method for predicting the asymptotic states of multistable systems using sparse measurements.
• To introduce a novel sparsity-promoting metric-learning (SPML) optimization for quantifying proximity to precomputed states.
• To ensure the learned metric is compatible with existing data and computable from sparse observations.

Main Methods:

• A semi-supervised classification approach was employed.
• Sparsity-Promoting Metric-Learning (SPML) optimization was introduced to learn a metric from precomputed data.
• The method was validated on a reaction-diffusion equation and a FitzHugh-Nagumo model.

Main Results:

• The SPML optimization was proven to be convex and non-degenerate.
• For a reaction-diffusion equation, SPML achieved 95% accuracy in predicting asymptotic behavior from two-point measurements.
• For the FitzHugh-Nagumo model, SPML achieved 90% accuracy from one-point measurements, also identifying optimal measurement locations.

Conclusions:

• The developed data-driven method effectively predicts asymptotic states of multistable systems using sparse measurements.
• SPML provides an accurate and efficient tool for analyzing complex systems in pattern formation and computational neuroscience.
• The learned metric guides efficient data acquisition for accurate state prediction.