Bounds on Extrapolation across Phase Transitions with Generalized Regression

Phase transitions occur when a physical system undergoes a dramatic transformation as it crosses a transition. We study whether regression methods trained only in one phase can predict the physical properties that characterize the unobserved phase. Our test sets consist of the analytically tractable Heisenberg model and angle-resolved photoemission spectroscopy measurements of a venerable high transition temperature superconductor. We find that polynomial regression and kernel methods recover the unobserved phase structure, while standard neural networks fail. To bound the error, we adopt the generalized eigenvalue problem (GEVP) into an upper bound on the worst-case transfer coefficient, defined as the ratio of mean squared error (MSE) loss on the extrapolation region to the MSE loss on the training region. Empirically, we find that our GEVP bound stays approximately constant with respect to the empirical transfer coefficient across training-set sizes and distance between the training and extrapolation region for both datasets, providing a principled framework for studying extrapolation feasibility.