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

2026-09-30

AI Summary

This study addresses the high computational cost of Monte Carlo simulations and the training instability caused by noisy labels in particle transport modeling. We propose directly training neural operators on noisy Monte Carlo data, theoretically demonstrating that unbiased noisy labels can substitute converged solutions for efficient training. Technically, we employ Softplus activations to achieve soft outputs in physical space and introduce a pointwise relative L2 loss (PRelL2) tailored for high-dynamic-range data, effectively overcoming high-variance challenges. Experiments show that this method achieves a 10,000-fold speedup in neutron transport tasks while significantly reducing computational costs with matched accuracy in radiative transfer applications.

Abstract

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 $L_2$ 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 $10^4$-$10^5\times$ faster than converged MC on the same CPU and $10^3$-$10^5\times$ cheaper than MC at matched accuracy; on the two radiative-transfer tasks, MC at matched accuracy costs $0.8$-$11\times$ as much as PTNO.

Problem

• Particle Transport
• Monte Carlo Simulation
• Neural Operator
• Noisy Labels
• High Dynamic Range

Innovation

• Neural Operator
• Monte Carlo
• Particle Transport
• High Dynamic Range
• Relative Loss