Automatic Generation of Sensitivity Systems for Differential-Algebraic Equations via Multivariate Taylor Series

We present a method for automatically generating sensitivity equations for parametrised differential-algebraic equations (DAEs) using multivariate Taylor se ries (MVTS) arithmetic with operator-overloading automatic differentiation. Given a DAE and a user-specified set of sensitivity orders (the sensitivity order vector, SOV), the method evaluates the original DAE once with MVTS-valued variables, extracts the coefficients, and assembles a self-contained extended complete sensi tivity system (ECSS)—an augmented DAE whose solution contains all requested sensitivities. The approach has three novel features: (i) the SOV framework with dependency closure enables selective computation of only the sensitivity orders the user needs, avoiding the combinatorial expansion of a full-order system; (ii) operator overloading MVTS arithmetic propagates all sensitivity orders simultaneously in a single evaluation pass, producing a structured DAE whose solvability is guar anteed by the structural inheritance theorem; and (iii) the generated ECSS is solver-independent—it is a plain DAE in standard form exportable to any integra tor accepting that interface. We demonstrate the method on the simple pendulum, showing how the coefficient arithmetic rules automatically produce the sensitivity equations, and report computational performance confirming linear cost scaling with the reliance number—a combinatorial metric derived from the SOV. Cross solver validation with SciPy’s Radau, RK45, and BDF integrators confirms that the generated equations are solver-independent, with agreement to 1.7 × 10−14 .