Machine learning surrogate models employing atomic environment descriptors have found wide applicability in materials science. In our previous work, this approach yielded accurate and transferable predictions of the vibrational formation entropy of point defects for O(N) computational cost. The present study investigates the limits of data driven surrogate models in accuracy and applicability for vibrational properties. We propose an improvement of the accuracy by extending the fitting capacity of the model by increasing the dimension of the descriptor space. This is achieved by using a nonlinear relation between descriptors—target observables and when it is possible by including physical relevant information of the underlying energy landscape. The nonlinear extension is used to learn the formation entropy of defects with or without applied strain while including physical information, such as the minimum-saddle point sequences employed for the migration of point defects, is a key ingredient of transition state theory rate approximations. We find excellent predictive power after augmenting the dimensionality of the descriptor space, as demonstrated on large defect databases in α-iron and amorphous silicon based on semiempirical force fields. The current linear surrogate models are used to investigate the correlation between migration entropy and energy. Our approaches reproduce the Meyer-Neldel compensation law observed from direct calculations in amorphous Si systems. Moreover, the same abstract descriptor space representation for entropy and energy is then used for the statistical correlation analysis. For linear surrogate models, we show that the energy-entropy statistical correlations can be reinterpreted in descriptor space. This provides a simple statistical criterion for the marginal interpretation of the compensation law. More generally, the present work shows how linear surrogate models can accelerate high-throughput workflows, aid the construction of mesoscale material models, and provide new avenues for correlation analysis.