  Key to being able to accurately model the properties of realistic materials
is being able to predict their properties in the thermodynamic limit.
Nevertheless, because most many-body electronic structure methods scale as a
high-order polynomial, or even exponentially, with system size, directly
simulating large systems in their thermodynamic limit rapidly becomes
computationally intractable. As a result, researchers typically estimate the
properties of large systems that approach the thermodynamic limit by
extrapolating the properties of smaller, computationally-accessible systems
based on relatively simple scaling expressions. In this work, we employ
Gaussian processes to more accurately and efficiently extrapolate many-body
simulations to their thermodynamic limit. We train our Gaussian processes on
Smooth Overlap of Atomic Positions (SOAP) descriptors to extrapolate the
energies of one-dimensional hydrogen chains obtained using two high-accuracy
many-body methods: Coupled Cluster theory and Auxiliary Field Quantum Monte
Carlo (AFQMC). In so doing, we show that Gaussian processes trained on
relatively short, 10-30-atom chains can predict the energies of both
homogeneous and inhomogeneous hydrogen chains in their thermodynamic limit with
sub-milliHartree accuracy. Unlike standard scaling expressions, our GPR-based
approach is highly generalizable given representative training data and is not
dependent on systems' geometries or dimensionality. This work highlights the
potential for machine learning to correct for the finite size effects that
routinely complicate the interpretation of finite size many-body simulations.
