Researchers have developed MANGO (MANGO: A Neutrino Gradient Oscillator), a new computational engine that allows for the calculation of neutrino oscillation probabilities and, crucially, their derivatives with respect to a wide range of parameters. While calculating oscillation probabilities is a solved problem, the ability to efficiently and precisely obtain their derivatives is a significant novelty. MANGO integrates automatic differentiation, meaning every computed quantity is differentiable with respect to all inputs, including propagation geometry, detector depth, and even individual Earth-shell densities, parameters not typically found in traditional analytical formulas.

This engine is versatile, supporting various propagation scenarios such as vacuum, constant density, layered Earth models (PREM), arbitrary profiles, and adiabatic solar propagation. Furthermore, it includes front-ends for non-standard interactions, 3+N sterile states, decoherence, and non-unitary mixing. MANGO's efficiency is remarkable: the cost of evaluating sensitivities is only 2.5 to 3 times that of a forward pass. This allows, for example, calculating sensitivities for all 369 density, electron-fraction, and shell-radius parameters of a layered Earth model at a cost comparable to that of the six standard oscillation parameters.

MANGO's ability to maintain exact sensitivity signals allows gradients to flow continuously through all stages of an analysis, from probability to detector response, event weighting, binning, and likelihoods. This has been demonstrated in a stylized Earth-tomography analysis, where MANGO computed the marginalized uncertainty on a six-zone radial density model and evaluated its sensitivity to detector angular resolution. This functionality provides an experimental design metric previously unreachable using traditional analytical probability formulas. MANGO's three-flavor and layered-Earth probabilities match external benchmarks (OscProb, NuFast-Earth) to within 10^-9 to 10^-5, and all forward models, BSM limits, and differentiation paths have been verified against exact analytical solutions and finite differences.