  We present GausSN, a Bayesian semi-parametric Gaussian Process (GP) model for
time-delay estimation with resolved systems of gravitationally lensed
supernovae (glSNe). GausSN models the underlying light curve non-parametrically
using a GP. Without assuming a template light curve for each SN type, GausSN
fits for the time delays of all images using data in any number of wavelength
filters simultaneously. We also introduce a novel time-varying magnification
model to capture the effects of microlensing alongside time-delay estimation.
In this analysis, we model the time-varying relative magnification as a sigmoid
function, as well as a constant for comparison to existing time-delay
estimation approaches. We demonstrate that GausSN provides robust time-delay
estimates for simulations of glSNe from the Nancy Grace Roman Space Telescope
and the Vera C. Rubin Observatory's Legacy Survey of Space and Time
(Rubin-LSST). We find that up to 43.6% of time-delay estimates from Roman and
52.9% from Rubin-LSST have fractional errors of less than 5%. We then apply
GausSN to SN Refsdal and find the time delay for the fifth image is consistent
with the original analysis, regardless of microlensing treatment. Therefore,
GausSN maintains the level of precision and accuracy achieved by existing
time-delay extraction methods with fewer assumptions about the underlying shape
of the light curve than template-based approaches, while incorporating
microlensing into the statistical error budget rather than requiring
post-processing to account for its systematic uncertainty. GausSN is scalable
for time-delay cosmography analyses given current projections of glSNe
discovery rates from Rubin-LSST and Roman.
