Models of hybrid dynamical systems are widely used to answer questions about the causes and effects of dynamic events in time. Unfortunately, existing causal reasoning formalisms lack support for queries involving the dynamically triggered, discontinuous interventions that characterize hybrid dynamical systems. This mismatch can lead to ad-hoc and error-prone causal analysis workflows in practice. To bridge the gap between the needs of hybrid systems users and current causal inference capabilities, we develop a rigorous counterfactual semantics by formalizing interventions as transformations to the constraints of hybrid systems. Unlike interventions in a typical structural causal model, however, interventions in hybrid systems can easily render the model ill-posed. Thus, we identify mild conditions under which our interventions maintain solution existence, uniqueness, and measurability by making explicit connections to established hybrid systems theory. To illustrate the utility of our framework, we formalize a number of canonical causal estimands and explore a case study on the probabilities of causation with applications to fishery management. Our work simultaneously expands the modeling possibilities available to causal inference practitioners and begins to unlock decades of causality research for users of hybrid systems.

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)

One approach is with hybrid systems, which are dynamical systems characterized by piecewise continuous trajectories with a finite number of discontinuities introduced by discrete events [5].

Notably, modern tools for the verification of hybrid automata are designed formodels thatrarely haveoverhundred discrete states [7],while arbitrary meshes grow exponentially as the granularity increases.

Since many areas, such as biology or discrete-event systems, are naturally described in continuous time, we present a model based on a Markov jumpprocessmodulating asubordinated diffusionprocess. Weprovidetheexact evolution equations fortheprior andposterior marginal densities, thedirect solutions of which are however computationally intractable.