Differentiable Parameter Optimization for DAEs with State-Dependent Events

TL;DR

This paper develops two gradient-based methods for optimizing parameters in differential-algebraic equations with state-dependent events, addressing challenges posed by implicit algebraic variables and discontinuities.

Contribution

It introduces an automatic-differentiation-through-simulation approach and an explicit discrete-adjoint method for differentiable parameter optimization in DAEs with events.

Findings

Both methods provide valid gradients under fixed event order and transversal guard crossings.

The approaches clarify the role of residual terms in the adjoint method as equality constraints.

Comparison of the two methods highlights their differences in interpretation and implementation.

Abstract

Differential-algebraic equations (DAEs) with state-dependent events arise in systems whose continuous dynamics are constrained by algebraic equations and interrupted by mode changes, switching logic, impacts, or state reinitializations. Gradient-based parameter learning for such systems is challenging because algebraic variables are implicitly defined, event times depend on the parameters, and reset maps introduce discontinuities. This paper studies differentiable parameter optimization for semi-explicit DAEs with events. We formulate the learning problem as a constrained least-squares problem with DAE dynamics, algebraic constraints, guard equations, and reset maps. We then develop two complementary gradient-computation strategies. The first is an automatic-differentiation-through-simulation method that solves algebraic variables inside the vector field, differentiates the algebraic solve using the implicit function theorem, and handles events through segmented differentiable integration. The second is an explicit discrete-adjoint method that represents the forward simulation as an event-split residual system and computes gradients by solving for the Lagrange multipliers of smooth-segment and event residuals. The formulation clarifies that residual terms in the adjoint method are equality constraints, not heuristic penalties. We compare the two approaches in terms of gradient interpretation, event-time handling, implementation complexity, and local validity. Both methods provide gradients for the event path selected by the forward simulation and are valid under fixed event ordering and transversal guard crossings.