Metal additive manufacturing (AM) typically suffers from high degrees of variability in the properties/performance of the fabricated parts, particularly due to the lack of understanding and control over the physical mechanisms that govern microstructure formation during fabrication. This paper directly addresses an important problem in metal AM: the determination of the thermal history of the deposited material. Any attempts to link process to microstructure in AM would need to consider the thermal history of the material. In situ monitoring only provides partial information and simulations may be necessary to have a comprehensive understanding of the thermo-physical conditions to which the deposited material is subjected. We address this in the present work through linking thermal models to experiments via a computationally efficient surrogate modeling approach based on multivariate Gaussian processes (MVGPs). The MVGPs are then used to calibrate the free parameters of the multi-physics models against experiments, sidestepping the use of prohibitively expensive Monte Carlo-based calibration. This framework thus makes it possible to efficiently evaluate the impact of varying process parameter inputs on the characteristics of the melt pool during AM. We demonstrate the framework on the calibration of a thermal model for laser powder bed fusion AM of Ti-6Al-4V against experiments carried out over a wide window in the process parameter space. While this work deals with problems related to AM, its applicability is wider as the proposed framework could potentially be used in many other ICME-based problems where it is essential to link expensive computational materials science models to available experimental data.