Reliable Transfer Learning for Quantum Phase Transitions

Predicting physical behavior beyond the data distribution is common in condensed matter physics but risky without guarantees. We study extrapolation of the mean-field free energy and critical temperature in a simplified Ising model using (i) Gaussian Processes (GPs), (ii) multivariate polynomial regression, and (iii) neural networks (MLPs). Building on recent transfer results for low-degree polynomials \citep{Kalavasis2024}, we compute explicit extrapolation bounds when training and test regions are disjoint in temperature. Empirically, degree-d polynomials trained only on the high-temperature phase extrapolate the free energy accurately into the low-temperature phase, with errors well below theoretical limits despite the bounds scaling as d^d. GPs provide the strongest accuracy but limited interpretability, and MLPs fit the training domain well yet degrade out-of-domain. Extending this analysis, we connect Carbery–Wright anti-concentration and Remez-type inequalities to more tightly bound how polynomial residuals can spread under domain shift, yielding refined L₁/L₂ extrapolation guarantees. These results indicate that low-degree polynomial predictors—unlike expressive black-box models—admit mathematically controlled transfer across phases. Our broader goal is to develop a unified framework for reliable transfer learning in physics: establishing provably small extrapolation errors for low-complexity models, validating these bounds across canonical phase-transition systems, and formulating a practical algorithmic recipe for experimentalists to extrapolate physical laws beyond observed regimes.