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 simulator's tangent dynamics and classical adjoint as the amplitude vanishes, while the learned observables carry only the finite-amplitude correction. On three nonlinear PDE systems, SALTO predicts the measurements of finite-amplitude sources to within 2-8% where the classical adjoint is in error by 41-165%, recovers source position and intensity from sparse sensors nearly as accurately as inversion through the solver, and samples the joint posterior of two interacting sources, matching the solver's posterior on reaction-diffusion at a twentieth of its cost. One trained model serves new sensor layouts and measurement times with a single backward solve.

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