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A large fraction of Type Ia supernova (SN Ia) observations over the nextdecade will be in the near-infrared (NIR), at wavelengths beyond the reach ofthe current standard light-curve model for SN Ia cosmology, SALT3 ($\sim2800$--8700$A$ central filter wavelength). To harness this new SN Ia sample andreduce future light-curve standardization systematic uncertainties, we trainSALT3 at NIR wavelengths (SALT3-NIR) up to 2 $\mu$m with the open-sourcemodel-training software SALTShaker, which can easily accommodate futureobservations. Using simulated data we show that the training process constrainsthe NIR model to $\sim 2$--3% across the phase range ($-20$ to $50$ days). Wefind that Hubble residual (HR) scatter is smaller using the NIR alone oroptical+NIR compared to optical alone, by up to $\sim 30$% depending on filterchoice (95% confidence). There is significant correlation between NIRlight-curve stretch measurements and luminosity, with stretch and colorcorrections often improving HR scatter by up to $\sim20%$. For SN Iaobservations expected from the \textit{Roman Space Telescope}, SALT3-NIRincreases the amount of usable data in the SALT framework by $\sim 20$% atredshift $z\lesssim0.4$ and by $\sim 50$% at $z\lesssim0.15$. The SALT3-NIRmodel is part of the open-source {\tt SNCosmo} and {\tt SNANA} SN Ia cosmologypackages.

SALT3-NIR: Taking the Open-Source Type Ia Supernova Model to Longer Wavelengths for Next-Generation Cosmological Measurements