# PTNO: Training Neural Operators with Noisy Monte Carlo Estimates for Particle Transport Problems

arXiv: https://arxiv.org/abs/2609.40090 · HTML: https://mlsanity.com/paper/2609.40090/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 53 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 21 of 21)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
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- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [NeRF in the Dark: High Dynamic Range View Synthesis from Noisy Raw Images](https://mlsanity.com/paper/2111.13679/) (2021)
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- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
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- [Deep Neural Operator Driven Real Time Inference for Nuclear Systems to Enable Digital Twin Solutions](https://mlsanity.com/paper/2308.07523/) (2023)
- [Prediction and Uncertainty Quantification of SAFARI-1 Axial Neutron Flux Profiles with Neural Networks](https://mlsanity.com/paper/2211.08654/) (2022)
- [Fast Regression of the Tritium Breeding Ratio in Fusion Reactors](https://mlsanity.com/paper/2104.04026/) (2021)
- [Operator Learning Using Weak Supervision from Walk-on-Spheres](https://mlsanity.com/paper/2603.01193/) (2026)
- [Nonlinear Noise2Noise for Efficient Monte Carlo Denoiser Training](https://mlsanity.com/paper/2512.24794/) (2025)
- [Learning Moment Maps for Continuous-Time Markov Chains under Monte Carlo Noise](https://mlsanity.com/paper/2606.18409/) (2026)

## Abstract (arXiv)

Particle transport under multiple scattering is central to radiative transfer and plasma physics, yet high-fidelity Monte Carlo (MC) simulations must trace prohibitively many particles. Learning-based surrogates can amortize this cost, but typically train on expensive, well-converged MC solutions. We propose the Particle Transport Neural Operator (PTNO), a neural operator that learns particle transport surrogates directly from noisy, low-cost MC labels. Such labels pose two challenges: (1) high variance, which destabilizes standard supervised learning, and (2) a high dynamic range (HDR) spanning many orders of magnitude. For the first, we learn the solution operator from noisy labels of many configurations, amortizing MC cost and generalizing to unseen configurations. Because MC labels are unbiased, we show that the squared loss on them shares its minimizer with the loss on converged solutions, and our budget-allocation study over training scenes M, MC samples per render N, and independent renders per scene K shows that many noisy scenes beat fewer converged ones. For the second, a nonlinear transform such as the logarithm biases noisy supervision. Instead, PTNO keeps labels in physical space and enforces positivity with a softplus output layer that represents small values effectively. We further train with a pointwise relative L2 loss (PRelL2), the stop-gradient relative loss of HDR denoising and neural rendering, which normalizes each residual by the stop-gradient prediction instead of the noisy label. We demonstrate PTNO on neutron transport in fusion reactors and radiative transfer in participating media. On the two neutronics tasks, PTNO is 104-105× faster than converged MC on the same CPU and 103-105× cheaper than MC at matched accuracy; on the two radiative-transfer tasks, MC at matched accuracy costs 0.8-11× as much as PTNO.