A Differentiable Neural Surrogate for Photon Propagation in Neutrino Telescopes

Technical Summary: A Differentiable Neural Surrogate for Photon Propagation in Neutrino Telescopes

Problem Statement
High-fidelity simulation of photon propagation is central to the scientific program of the IceCube Neutrino Observatory, a cubic-kilometer detector embedded in Antarctic glacial ice. The detector infers neutrino properties from Cherenkov light produced by charged particles. However, simulating the transport of billions of photons through the highly scattering and absorbing, heterogeneous ice is computationally prohibitive. Explicit Monte-Carlo (MC) propagation, such as the Photon Propagation Code (PPC), has become a dominant bottleneck in the simulation pipeline; a single 2019 campaign consumed approximately 100,000 GPU-hours for photon tracking alone.

This computational burden is exacerbated by IceCube's extreme event-rate imbalance. While atmospheric muons trigger the detector at kHz rates, the rare astrophysical neutrino signals occur at mHz or µHz rates. Accurately estimating the background rejection requires generating enormous simulated samples of atmospheric muons. Furthermore, the ice medium introduces substantial complexity, including depth-dependent optical properties, anisotropic propagation, and tilted ice layers, requiring simulations to reproduce not only total light yield but also nanosecond-scale arrival-time distributions. Existing methods like Photonics (lookup tables) or photospline (spline fits) struggle with the dimensionality of the problem or require dense MC tables for every new medium model.

Methodology: The candela Framework
The authors introduce candela (Complex-media-Aware Network for Differentiable and Efficient Light-transport Approximation), a differentiable neural surrogate designed to learn the photon Green's function of the IceCube detector. The approach relies on the linearity of photon transport with respect to emitted light, allowing complete events to be decomposed into localized energy deposits.

1. Source-Receiver Decomposition:

   • A complete event (e.g., a muon track) is decomposed into a sequence of point-like cascade sources.
   • The simulation is reduced to evaluating a shared source-receiver kernel for each pair of a light source and a sensor.
   • The input query ($x_{sr}$) includes the relative geometry in a cylindrical frame aligned with the source direction ($R, \Phi, l$), the source direction and energy, and the absolute positions of the source and receiver (to account for the depth-dependent optical properties of the ice).

2. Neural Field Architecture:

   • The core is a SIREN (Sinusoidal Representation Network) based implicit neural representation with six layers and a width of 512.
   • The network predicts two outputs for a given query:
     • $\Lambda(x_{sr})$: The expected detected photon yield per unit deposited energy.
     • $\pi(\tau | x_{sr})$: A unit-normalized residual arrival-time density, modeled as an eight-component mixture of inverse-Gaussian distributions.
   • The arrival time is expressed as a residual $\tau = t - \Delta t_{sr}$, where $\Delta t_{sr}$ accounts for the emission time and unscattered geometric time-of-flight. This allows the network to focus on modeling delays and broadening induced by scattering.

3. Event Composition:

   • For a full event, the expected photon rate at a receiver $r$ is computed by summing the contributions of all sources $s$, scaled by their deposited energy $E_s$:
     $$\mu_r(t) = \sum_s E_s G(t - \Delta t_{sr} | x_{sr})$$
   • This composition allows for parallel evaluation of all source-receiver pairs.

4. Training:

   • The model is trained on data generated by the PPC simulator using 65,536 isolated electromagnetic cascade configurations.
   • Training targets a fixed total number of produced photons, with repetition counts varying by source energy.
   • The loss function is a Poisson likelihood over sparse histograms of spatial bins ($R, \Phi, l$) and residual time bins, jointly constraining the integrated yield and the timing distribution.

Key Contributions

• Differentiable Surrogate: candela provides an end-to-end differentiable forward model, enabling gradient-based optimization with respect to event parameters (position, direction, energy) and, potentially, medium properties.
• Efficiency: By replacing individual photon tracking with batched evaluations of the learned kernel, the method decouples computational cost from photon multiplicity, making it dependent primarily on event geometry.
• Generalizability: The source-receiver formulation is designed to be extensible to other detectors and scientific instruments requiring efficient photon transport through complex scattering media.

Results
Evaluated on independent neutrino events spanning $10^2$ to $10^6$ GeV, candela demonstrates the following:

• Speed: The model generates events 50–100× faster than the standard PPC GPU-based MC simulator. While PPC time scales with the number of photons (taking ~1 second at low energy and increasing with energy), candela requires only 10–20 ms with weak dependence on neutrino energy.
• Yield Accuracy: The median predicted photon yield remains within 2% of the high-statistics MC expectation across six orders of magnitude in photon count.
• Timing Fidelity: The predicted arrival-time distributions match the MC statistical floor (the irreducible error from finite photon statistics) across the same range. The Kolmogorov–Smirnov (KS) distance between candela predictions and high-statistics PPC histograms tracks the PPC-PPC variance, except in the regime of photomultiplier-tube (PMT) saturation, where the model begins to separate from the statistical floor.
• Event-Level Agreement: Visual and quantitative comparisons show that candela reproduces spatial light deposition patterns, prompt peaks, scattering tails, and first-hit time progressions accurately.

Significance and Claims
The paper claims that candela provides a practical foundation for accelerating simulation in large-volume neutrino telescopes without sacrificing fidelity. By reproducing photon yields and timing distributions to within the MC statistical limit, it enables the generation of the massive background samples required for rare signal extraction.

Crucially, the differentiability of the surrogate opens a path toward optimizing scattering-medium properties, which often dominate systematic uncertainties in neutrino telescopes. While the current study focuses on establishing efficiency and fidelity, the authors note that the framework lays the groundwork for future gradient-based event reconstruction and detector calibration. The authors modestly acknowledge that the model's performance degrades near sensor saturation (a regime often ignored in practice) and that future work is needed to fully leverage the differentiability for reconstruction tasks and to condition the network on varying optical parameters for broader applicability.