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MANGO: An Autodiff Neutrino Oscillation Engine for Differentiable Analysis Pipelines

This paper introduces MANGO, a fully differentiable neutrino oscillation engine that enables efficient, end-to-end gradient-based analysis of complex propagation scenarios and experimental designs by computing sensitivities for all inputs—including Earth density profiles and detector parameters—with minimal computational overhead.

Original authors: Pierre Granger

Published 2026-08-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Pierre Granger

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Invisible Ghosts and the Math That Moves Them

Imagine the universe is filled with tiny, ghostly particles called neutrinos. They are so shy that they can pass through a light-year of solid lead without bumping into anything, and trillions of them are zipping through your body right now, completely unnoticed. These particles come in three "flavors" (think of them like different colors: electron, muon, and tau), and as they travel across the vast distances of space or through the Earth, they have a magical habit of changing their color. This phenomenon is called "neutrino oscillation."

Scientists have been studying these shape-shifters for decades to understand the fundamental rules of the universe. To do this, they build massive detectors deep underground and wait for a neutrino to interact. But to figure out what the neutrino was doing, they need to run complex computer simulations. These simulations act like a time machine, predicting how a neutrino should behave as it travels through the Earth's core, the mantle, and the atmosphere.

For a long time, these simulations were like a one-way street. You could put in the starting conditions (where the neutrino came from, how fast it was going) and get an answer (what flavor it is when it arrives). But if you wanted to know why the answer changed when you tweaked a single number—like the density of the Earth's core or the angle at which the neutrino entered—you had to run the whole simulation again, change the number slightly, and run it again. It was like trying to find the perfect recipe for a cake by baking a new one every time you wanted to know if adding a pinch more sugar would help. It worked, but it was slow, clumsy, and made it nearly impossible to optimize the experiment itself.

The New Engine: MANGO

Enter MANGO (which stands for "MANGO: A Neutrino Gradient Oscillator"). Think of MANGO not just as a calculator, but as a super-smart, self-aware engine that can do the math in reverse. In the world of computer science, this is called "automatic differentiation." While traditional tools can only tell you the result of a calculation, MANGO can tell you exactly how sensitive that result is to every single ingredient used to make it, all in one go.

The paper introduces MANGO as a new, open-source tool built on a modern programming framework called JAX. Its main superpower is that it treats the entire process of neutrino physics as a single, continuous chain of math. Usually, scientists calculate the probability of a neutrino changing flavors, and then stop. MANGO keeps the "math chain" open. It allows the computer to trace the sensitivity of the final result all the way back through the detector's response, the way the data is grouped, and even the physical shape of the Earth itself.

What MANGO Actually Does

The researcher built MANGO to solve a specific headache: calculating derivatives (slopes) of neutrino probabilities. In the past, computing these slopes was either impossible or required scientists to manually write out complex, error-prone formulas for every single new scenario. MANGO changes the game by making every single step of the calculation differentiable.

Here is what the paper demonstrates:

It sees the invisible: MANGO can calculate how the neutrino probability changes if you tweak things that traditional formulas ignore, like the exact depth of the detector, the height where the neutrino was created in the atmosphere, or the density of specific layers inside the Earth.
It handles the messy stuff: The Earth isn't a perfect sphere with smooth layers; it has a crust, a mantle, and a core, each with different densities. MANGO can model this "layered" Earth (using a standard model called PREM) and even handle weird, non-standard physics theories where neutrinos might interact with matter in new ways or lose their quantum "coherence."
It's incredibly fast: The paper shows that calculating the sensitivity for all 369 different parameters of a layered Earth model (densities, electron fractions, and shell radii) takes only about 2.5 to 3 times longer than running the simulation once. In the old way, doing this would have required hundreds of separate runs.
It's precise: The author verified that MANGO's results match other established tools (like OscProb and NuFast) to within a tiny fraction of a percent (specifically, differences as small as $10^{-9}$ to $10^{-5}$ ). They also proved that the math works perfectly even when neutrinos pass through the Earth's core, a region where the math usually gets very tricky.

A Real-World Test: The Earth Tomography

To show off its power, the author used MANGO to perform a "tomographic" analysis of the Earth. Imagine trying to figure out the density of the Earth's core by watching neutrinos pass through it.

Using MANGO, they didn't just guess the density; they calculated exactly how much uncertainty there would be in that measurement. But here is the kicker: they then differentiated through that uncertainty to see how it would change if they improved the detector's angular resolution (how well the detector can tell the direction of the neutrino).

In a single pass, MANGO calculated that improving the detector's ability to measure angles by just 0.01 would improve the precision of the Earth's core density measurement by 7.8%. This is a type of "experimental design" calculation that was previously unreachable because it required differentiating through the entire chain of physics, statistics, and detector hardware.

What It's Not (and What It Doesn't Do)

It is important to note what MANGO is not. It is not a tool for simulating high-energy neutrinos that get absorbed or destroyed by the Earth (a process called attenuation); for that, other tools are still needed. It is also not a magic wand that solves every physics problem instantly; it is a foundation. The author explicitly states that while MANGO handles the core oscillation physics perfectly, extending it to cover full density-matrix transport or complete experiment simulations (like those done by other large software suites) is work for the future.

The Bottom Line

The paper presents MANGO as a foundational shift. It moves neutrino physics from a world where you calculate a result and then manually guess how to improve it, to a world where the computer tells you exactly how to tweak the experiment, the detector, or the model to get the best possible answer. It turns the "black box" of neutrino simulation into a transparent, adjustable machine, allowing scientists to optimize their experiments and understand the Earth's interior with a level of precision and speed that was previously out of reach. The author has made the code open-source, inviting the rest of the scientific community to build on this new, differentiable foundation.

Technical Summary: MANGO – An Autodiff Neutrino Oscillation Engine for Differentiable Analysis Pipelines

Problem Statement
While computing neutrino oscillation probabilities is a well-established problem with mature, highly optimized software (e.g., Prob3++, OscProb, GLoBES, NuFast), computing their exact derivatives and embedding them into end-to-end differentiable analysis pipelines remains an unaddressed challenge. Existing tools excel at forward probability evaluations but lack automated, exact derivatives. This architectural limitation prevents the integration of oscillation physics into modern differentiable workflows, such as Hamiltonian Monte Carlo (HMC), variational inference, neural surrogates, and gradient-based experimental design. Furthermore, analytic derivative formulas are often restricted to specific regimes (e.g., constant density) and cannot easily accommodate complex geometries (like layered Earth models) or non-standard parameters (such as detector depth or atmospheric production height) that do not appear in closed-form probability expressions.

Methodology
The paper presents mango (MANGO: A Neutrino Gradient Oscillator), a composable oscillation engine implemented natively in JAX. Rather than deriving analytic derivatives for specific cases, mango implements a general propagation core and relies on Automatic Differentiation (AD) to obtain exact derivatives with respect to all inputs.

Propagation Core: The engine supports vacuum, constant-density, layered-PREM, arbitrary-profile, and adiabatic solar propagation. It handles standard three-flavor oscillations as well as Beyond-Standard-Model (BSM) scenarios, including non-standard interactions (NSI), 3+N sterile states, environmental decoherence, and non-unitary mixing.
Geometry and Differentiability: The framework treats propagation geometry (zenith angle, production height, detector depth) as differentiable inputs. By constructing the trajectory chord and shell intersections via closed-form differentiable functions, the engine computes derivatives with respect to geometric parameters that are absent in traditional analytic formulas.
Numerical Backend: To ensure numerical precision and differentiability across spectral degeneracies, mango employs a divided-difference expansion (Cayley–Hamilton) as its default production engine. This approach avoids explicit eigenvector computation, thereby bypassing the derivative singularities ( $1/(\lambda_i - \lambda_j)$ ) that plague standard diagonalization routines near mass-level crossings. Alternative backends include a JAX-transcribed NuFast algorithm, dense Hermitian diagonalization, and Padé exponentials for verification.
Differentiability Model: The engine supports both forward-mode and reverse-mode AD. Crucially, the reverse-mode cost depends on the output dimension rather than the input dimension. This allows the evaluation of sensitivities for hundreds of parameters (e.g., 369 parameters for a layered Earth model) with a constant overhead of only 2.5–3× a single forward pass.

Key Contributions

Composable Differentiable Engine: mango provides the first oscillation engine where every computed quantity is differentiable with respect to all inputs, including propagation geometry, detector depth, and individual Earth-shell densities.
Coverage and Reach: The engine supports derivatives for every configuration it handles (layered PREM, continuous solar profiles, NSI, sterile states). It enables differentiation with respect to quantities not present in analytic formulas, such as the chord geometry parameters ( $\cos \theta_z$ , $h_{atm}$ , $d_{det}$ ) and per-shell densities.
End-to-End Differentiability: The AD graph does not terminate at the probability calculation. It flows continuously through detector response, event weighting, histogram binning, and likelihoods. This allows a single jax.grad call to return the gradient of a physics-analysis figure of merit with respect to any upstream quantity.
Numerical Robustness: By using divided differences, the engine guarantees $C^\infty$ smooth AD derivatives across exact mass level-crossings and MSW resonances, avoiding the numerical instabilities associated with eigenvector differentiation.

Results

Accuracy: Three-flavor and layered-Earth probabilities match external benchmarks (OscProb, NuFast-Earth) to within $10^{-9}$ to $10^{-5}$ . All forward models, BSM limits, and differentiation paths are verified against exact analytic solutions and finite differences, with relative deviations typically in the range of $10^{-10}$ to $10^{-5}$ .
Performance: On an NVIDIA A100 GPU, the constant-density NuFast port runs in 61 ns per energy point, while the general divided-difference kernel runs in 116 ns. The layered PREM path (involving ~50 segments) costs approximately 158 $\mu$ s per point.
Scaling: Evaluating sensitivities for all 369 density, electron-fraction, and shell-radius parameters of a layered Earth costs no more than the 6 standard oscillation parameters (a cost factor of ~3.0× forward pass). In contrast, finite differencing would require ~250 times more evaluations.
Demonstration: The paper demonstrates a stylized Earth-tomography analysis. In a single pass, mango computes the marginalized uncertainty on a six-zone radial density model. By differentiating through the inverse Fisher matrix, it evaluates the sensitivity of this uncertainty with respect to detector angular resolution. This yields an experimental-design metric (e.g., how much core-density precision improves with better angular resolution) that is unreachable using traditional analytic probability formulas.

Significance
The paper claims that mango provides a foundational shift for neutrino oscillation workflows, moving from traditional forward-only calculators to an exact, end-to-end differentiable engine. Its primary significance lies in enabling differentiable inference and experimental design where physics models, instrument design parameters, and analysis pipelines can be jointly differentiated and optimized. By maintaining exact sensitivity signals through the entire pipeline, mango allows for:

Experimental Design: Optimizing binning schemes and evaluating hardware trade-offs (e.g., angular resolution vs. tomography reach) via gradient descent.
High-Dimensional Fits: Enabling fast marginalization over hundreds of systematic nuisance parameters using gradient-based minimizers and HMC.
Geophysical Inversions: Facilitating gradient-based inversion techniques to reconstruct continuous Earth interior density profiles directly from neutrino data.

The work establishes a composable, differentiable foundation that connects fundamental quantum mechanics directly to experimental discovery potential, addressing a critical bottleneck in modern high-energy physics analysis.

The Invisible Ghosts and the Math That Moves Them

The New Engine: MANGO

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