Optimal criteria and their asymptotic form for data selection in data-driven reduced-order modelling with Gaussian...

Area of Science:

• Computational Mathematics
• Machine Learning
• Dynamical Systems

Background:

• Data-driven reduced-order modeling (ROM) and supervised learning often rely on Gaussian Process Regression (GPR).
• Current data selection criteria in active learning and optimal experimental design are largely empirical.
• Efficient data selection is crucial for accurate and robust GPR-based models.

Purpose of the Study:

• To derive theoretically grounded criteria for selecting optimal datapoints for GPR-based data-driven reduced-order modeling.
• To address the limitations of empirical data selection methods.
• To enhance the convergence and efficiency of GPR models in supervised learning tasks.

Main Methods:

• Formulated an optimality condition based on minimizing the distance between approximated and exact output probability density functions (pdfs).
• Defined a selection criterion using the supremum over the native Hilbert space of GPR.
• Employed GPR theory and asymptotic analysis to derive a computable form of the criterion for small predictive variance.

Main Results:

• Developed a computable selection criterion for GPR datapoint selection.
• The criterion ensures balanced data resource distribution between probable and large-deviation outputs.
• Demonstrated convergence of the GPR model through this balanced sampling approach.

Conclusions:

• The derived criterion offers a principled and effective method for data selection in GPR for ROM.
• This approach improves model convergence and resource allocation.
• Contributes to advancing data-driven prediction in dynamical systems.