Recent advances in frontal ring-opening metathesis polymerization (FROMP) offer a sustainable and energy-efficient alternative for rapid curing of thermoset polymers compared to conventional bulk curing. To predict FROMP dynamics for different formulations and processing conditions, we require an accurate continuum model. The driving force for FROMP lies in the underlying cure kinetics, but our understanding of the mechanisms is limited and existing cure kinetics models fall short. Herein, we demonstrate that a differentiable simulator for partial differential equations (PDEs) enables learning of cure kinetics functions from video frames of the true solution. With a hybrid PDE solver, where learnable terms are parameterized by orthogonal polynomials or neural networks, we can uncover missing physics within the PDE by applying PDE-constrained optimization and the adjoint method. Our work paves the way for learning spatiotemporal physics and kinetics from experimentally captured videos.