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Abstract

Modern supervised machine learning (ML) techniques have taken a prominent role in academia and industry due to their powerful predictive capabilities. While many large-scale ML models utilize deep artificial neural networks (ANNs), which have shown great success if large amounts of data are provided, ML methods employing Gaussian processes (GPs) outperform ANNs in cases with sparse training data due to their interpretability, resilience to overfitting, and provision of reliable uncertainty measures. GPs have already been successfully applied to pattern discovery and extrapolation. The latter can be done in a controlled manner due to their small numbers of interpretable hyperparameters. In this work we develop an approach based on GPs to extract diabatic patterns from energy spectra, adiabatic under variation of a parameter of the Hamiltonian. The emerging diabatic manifolds (or energy surfaces) exhibit crossings where the original (adiabatic) energy spectra avoid to cross. In the context of highly excited, classically chaotic dynamics, we demonstrate that our GP regression approach can generate complete diabatic energy spectra with two exemplary systems: two coupled Morse oscillators and hydrogen in a magnetic field. For both we train GPs with few classical trajectories in order to inter- and extrapolate actions throughout the whole energy and parameter range to identify all points where the semiclassical Einstein-Brillouin-Keller (EBK) quantization condition is fulfilled. While the direct EBK method is restricted to regular classical dynamics, the interpretability of the GPs allow for controlled extrapolation into regions where no more regular trajectories exist due to irregular motion. Hence, semiclassical diabatic spectra can be continued into chaotic regions, where such manifolds are no longer well-defined. Further, we investigate the origin of resonant motion in the coupled Morse oscillator system and their contributions to the semiclassical spectra, which provide energies along strongly repelled adiabatic surfaces. For the hydrogen atom in a magnetic field we show that a proper scaling of the coordinates by the magnetic field strength allows for the extraction of an infinite series of semiclassical energies with one single trajectory which fulfills the EBK condition. The implementation of boundary conditions into GPs, as well as scaling techniques to higher dimensions and their properties are discussed.