A.1 Neural Hybrid Automata: Modules and Hyperparameters We provide a notation and summary table for Neural Hybrid Automata (NHA). The table serves as a quick reference for the core concepts introduced in the main text. Labels every subjtrajectory Xi with a mode z to ensure mode-conditioned decoder Fz can reconstruct it despite Neural ODE representation limitations (uniqueness of solutions given an initial condition). The only NHA hyperparameter beyond module architectural choices is m, or number of latent modes provided to the model at initialization. Performance effects of changing mhave been explored in Section 5.2 and Appendix B.2. Appendix B.2 further provides analyzes potential techniques to prune additional modes. A.2 Gradient Pathologies We provide some theoretical insights on the phenomenon of gradient pathologies with the simple example of a one-dimensional linear hybrid system with two modes and one timed jump, xt = axtt<τ bxtt>= τ t 6= τ x+t = cxtt= τ (A.1)

Intensity,as defined in the classical temporal point process(TPP) sense, can be interpreted as the expected number of eventsz z0 withinthetimeinterval[t,t+dt].