SALTO: Solver-Anchored Lifted Transfer Operators for Nonlinear Event Inference

Qi Wang, Rylie Foster

Learned dynamical models can accurately predict trajectories while producing unreliable sensitivities when differentiated or transposed, limiting their use in inverse problems and scientific decision-making. Classical adjoints provide exact local derivatives, but a single adjoint cannot represent the nonlinear response of a system to a family of finite-amplitude perturbations. We introduce SALTO (Solver-Anchored Lifted Transfer Operators), a Koopman-based framework for inferring finite-amplitude sources. In the lifted space, the nonlinear dynamics are approximated by a linear propagator, and its transpose carries each sensor backward in a single solve. The result is a reusable lifted adjoint that predicts the measurement produced by any candidate localized event with one encoder pass and an inner product. To give the model the simulator's exact local behavior, SALTO imposes three structural constraints: it anchors the physical-state dynamics to the simulator's linearized propagator, centers the learned observables so that they vanish to second order at the reference state, and uses a block-triangular lifted propagator. These constraints guarantee that the model reduces to the…