In recent years, the construction of data-driven force fields via Machine Learning methods proved to be a promising route in order to bridge the gap between accurate (but slow) quantum calculations and fast (but unreliable) classical potentials [1,2,3].

I will present a new scheme [4] that accurately predicts forces as vector quantities, rather than sets of scalar components, by Gaussian Process (GP) Regression. This is based on matrix-valued kernel functions, to which we impose that the predicted force rotates with the target configuration and is independent of any rotations applied to the configuration database entries. We show that such "covariant" GP kernels can be obtained by integration over the elements of the rotation group SO(n). Remarkably, in specific cases the integration can be carried out analytically and yields a conservative force field that can be recast into a pair interaction form. The accuracy of our kernels in predicting quantum-mechanical forces in real systems is investigated by tests on pure and defective crystalline systems.

I will further discuss how such learning algorithm can be used to build a measure of complexity of physical systems. Indeed, this can be­ defined as the number of canonically sampled configurations needed to achieve low generalization error with high probability.

[1] Behler et al. M. (2007). Generalized Neural-Network Representation of High-Dimensional Potential-Energy Surfaces. Physical Review Letters. http://doi.org/10.1103/PhysRevLett.98.146401

[2] Bartók et al. (2010). Gaussian Approximation Potentials: The Accuracy of Quantum Mechanics, without the Electrons. Physical Review Letters. http://doi.org/10.1103/PhysRevLett.104.136403

[3] Li et al. (2015). Molecular Dynamics with On-the-Fly Machine Learning of Quantum-Mechanical Forces. Physical Review Letters. http://doi.org/10.1103/PhysRevLett.114.096405

[4] Glielmo et al. (under review). Accurate Force Fields via Machine Learning with Covariant Kernels. Physical Review B. https://arxiv.org/abs/1611.03877