A Performance Metric for Objective Discovery of Leading-Order Dynamics through Unsupervised Machine Learning

Unsupervised learning algorithms have recently been applied to equation space data (data for which the features are equation terms) to discover leading-order equation terms across non-asymptotic regimes. However, the tasks of choosing optimal unsupervised learning algorithm free parameters and of interpreting the resulting clusters of data have been reliant on human discretion and intuition. To fully objectify and automate leading-order equation term discovery, we introduce a performance metric that distinguishes favorable and unfavorable results by comparing the sums of the logs of absolute relative differences between equation terms. By comparing the leading-order balances (subsets of equation terms) to the full balances (all equation terms), the performance metric is an approximate measure of how successfully an algorithm-chosen leading-order balance leads the following-order terms. In this presentation, we demonstrate how this performance metric can be used to a) objectively determine the optimal choice of unsupervised learning algorithm free parameters, b) objectively discover if leading-order balances exist within equation space data, and c) objectively discover the optimal leading-order interpretation of clusters of data within equation space.