When investigating stochastic ignition phenomena, high-fidelity simulations and experiments for statistical characterization can be a laborious endeavor. Machine learning (ML), in particular deep-learning, offers an accurate and cost-effective approach for reduced-order modeling of forced ignition. In this work, we investigate the potential and limitations of ML in modeling ignition, especially in the presence of limited data and multiple ignition modes. To this end, we introduce a hybrid stochastic physics-embedded deep-learning framework that combines sparse experimental data from Schlieren measurements with inert large-eddy simulations for predicting the spatio-temporal evolution of ignition kernel location and morphology. This model combines a stochastic differential equation for modeling kernel dynamics and a deep-learning model for representing kernel morphology. We evaluate this modeling approach for laser ignition of a gaseous methane/oxygen (CH4/O2) model rocket combustor, where morphological effects of the ignition kernel significantly influence ignition behavior. Results demonstrate that this model can reasonably capture behavior associated with the three dominant ignition modes, namely direct, indirect, and failed ignition, along with statistics associated with kernel growth and position, at lower computational costs than high-fidelity reacting simulations. In addition, we demonstrate that this modeling framework can be employed for generating spatially resolved ignition probability maps by incorporating physics to represent kernel interaction with the turbulent jet. We note that limitations in accuracy can be observed when predicting with vastly out-of-distribution data. Nevertheless, these results demonstrate that this physics-embedded ML approach can statistically characterize forced ignition in a more cost-effective manner than reacting high-fidelity simulations, as long as sufficiently representative data is available.