# Physical extrapolation of quantum observables by generalization with Gaussian Processes

## TL;DR

This paper demonstrates that Gaussian processes can be used to accurately predict quantum properties outside the training data range, enabling physical extrapolation across quantum phase transitions with models of varying complexity.

## Contribution

The work introduces a method for physical extrapolation using Gaussian processes, including kernel complexity control with BIC, to predict quantum properties beyond training data.

## Key findings

- GP models can predict quantum phase transitions.
- Simple kernels suffice for analytic, simple physical functions.
- Complex kernels improve predictions for intricate physical systems.

## Abstract

For applications in chemistry and physics, machine learning (ML) is generally used to solve one of three problems: interpolation, classification or clustering. These problems use information about physical systems in a certain range of parameters or variables in order to make predictions at unknown values of these variables within the same range. The present work illustrates the application of ML to prediction of physical properties outside the range of the training parameters. We define `physical extrapolation' to refer to accurate predictions $y(\mathbf{x^\ast})$ of a given physical property at a point $\mathbf{x^\ast} = [ x^\ast_1, ..., x^\ast_{\cal D} ]$ in the $\cal D$-dimensional space, if, at least, one of the variables $x^\ast_i \in \left [ x^\ast_1, ..., x^\ast_{\cal D} \right ]$ is {\it outside} of the range covering the training data. We show that Gaussian processes (GPs) can be used to build ML models capable of physical extrapolation of quantum properties of complex systems across quantum phase transitions. The approach is based on training GP models of variable complexity by the evolution of the physical functions. We show that, as the complexity of the models increases, they become capable of predicting new transitions. We also show that, where the evolution of the physical functions is analytic and relatively simple, GP models with simple kernels already yield accurate generalization results, allowing for accurate predictions of quantum properties in a different quantum phase. For more complex problems, it is necessary to build models with complex kernels. The complexity of the kernels is increased using the Bayesian Information Criterion (BIC). We illustrate the importance of the BIC by comparing the results with random kernels of various complexity and illustrate a method to obtain meaningful extrapolation results without direct validation in the extrapolated region.