This paper introduces MANGO (A Neutrino Gradient Oscillator), a software engine that computes neutrino oscillation probabilities and, crucially, their derivatives with respect to many inputs. Computing the probabilities is a solved problem. Getting exact, automatic derivatives has been hard. MANGO fills that gap so gradients are available alongside the usual probability outputs.

The code is written in JAX and uses automatic differentiation. That means the derivatives are produced programmatically instead of being worked out by hand for each special case. MANGO supports many propagation regimes: vacuum, constant-density layers, a layered PREM (Preliminary Reference Earth Model) description of the Earth, arbitrary radial profiles, and an adiabatic model for solar propagation. It also has front ends for non-standard interactions, 3+N sterile states, decoherence, and non-unitary mixing. Geometry quantities such as the zenith angle, production altitude, and detector depth enter through a closed-form chord construction, so derivatives like ∂P/∂cosθz exist and are exact except at a set of special grazing angles.

A practical advantage is in cost. Using reverse-mode automatic differentiation (the method that gives gradients efficiently when there are many inputs), the authors report a constant 2.5–3× overhead compared with a forward probability calculation. Because that cost depends on the number of outputs rather than inputs, MANGO can compute sensitivities for all 369 layered-Earth parameters (densities, electron fractions, shell radii) at no greater expense than computing gradients for the six standard oscillation parameters.

Why this matters: many modern analysis tools and experiment-design methods need derivatives that flow through the whole pipeline — from oscillation probabilities via detector response and event weighting to binning and likelihoods. The authors demonstrate this by running a stylized Earth-tomography study. In a single pass MANGO computes the marginalized uncertainty on a six-zone radial density model and, by differentiating through the inverse Fisher matrix, evaluates how that uncertainty changes with detector angular resolution. The paper argues this yields experimental-design metrics that are hard or impossible to get with traditional analytic probability formulas.