Cost function for low-dimensional manifold topology assessment.

Area of Science:

• Computational Science
• Data Science
• Applied Mathematics

Background:

• Reduced-order modeling simplifies complex systems by projecting high-dimensional data onto lower-dimensional manifolds.
• The topology of these manifolds is critical for the accuracy of subsequent models, like nonlinear regression.
• Current methods lack robust ways to quantify and optimize manifold topology.

Purpose of the Study:

• To develop a quantitative metric for characterizing manifold topologies in reduced-order modeling.
• To demonstrate the use of this metric as a cost function for optimizing low-dimensional projections.
• To assess the impact of manifold quality on downstream modeling tasks.

Main Methods:

• A novel quantitative metric is proposed to assess manifold topology, considering non-uniqueness and spatial gradients.
• The metric is employed as a cost function within optimization algorithms to find optimized low-dimensional projections.
• The metric's efficacy is tested on diverse datasets including argon plasma, reacting flows, and atmospheric pollutant dispersion.

Main Results:

• Optimized low-dimensional projections were successfully identified using the proposed metric as a cost function.
• The metric effectively evaluates various dimensionality reduction and manifold learning techniques.
• Improved manifold topologies were shown to enhance the performance of nonlinear regression models.

Conclusions:

• The developed metric provides a robust tool for quantifying and optimizing manifold topology in reduced-order modeling.
• This approach facilitates the selection of superior dimensionality reduction strategies and data preprocessing techniques.
• Enhanced manifold quality directly translates to improved predictive power in complex system modeling.