Generative Monte Carlo Sampling for Constant-Cost Particle Transport

TL;DR

Generative Monte Carlo is presented, a novel paradigm for particle transport simulation that integrates generative artificial intelligence directly into the stochastic solution of the linear Boltzmann equation, and preserves the statistical fidelity of standard Monte Carlo.

Abstract

We present Generative Monte Carlo (GMC), a novel paradigm for particle transport simulation that integrates generative artificial intelligence directly into the stochastic solution of the linear Boltzmann equation. By reformulating the cell-transmission problem as a conditional generation task, we train neural networks using conditional flow matching to sample particle exit states, including position, direction, and path length, without simulating scattering histories. The method employs optical coordinate scaling, enabling a single trained model to generalize across any material. We validate GMC on two canonical benchmarks, namely a heterogeneous lattice problem characteristic of nuclear reactor cores and a linearized hohlraum geometry representative of high-energy density radiative transfer. Results demonstrate that GMC preserves the statistical fidelity of standard Monte Carlo, exhibiting the expected $1/\sqrt{N}$ convergence rate while maintaining accurate scalar flux profiles. While standard Monte Carlo computational cost scales linearly with optical thickness in the diffusive limit, GMC achieves constant $O(1)$ cost per cell transmission, yielding order-of-magnitude speedups in optically thick regimes. This framework strategically aligns particle transport with modern computing architectures optimized for neural network inference, positioning transport codes to leverage ongoing advances in AI hardware and algorithms.

Figures (4)

• Figure 1: GMC workflow and representative trajectories. Figure \ref{['fig:nf_workflow']} illustrates the transport pipeline. The conditioning vector $\mathbf{c} = (\tilde{W}, \tilde{H}, \bm{\xi}_{\text{in}}, \mathbf{\Omega}_{\text{in}})$ encodes optical dimensions, entry position, and direction; the output $\mathbf{y} = (p_{\mathrm{exit}}, \mathbf{\Omega}_{\text{exit}}, s)$ encodes exit position, direction, and path length. The workflow proceeds through four stages: (1) Internal (blue) and boundary (red) particles are initialized. (2) Separate networks process each type; boundary particles are rotated into the canonical left-entry frame. (3) Particles are mapped to global coordinates and energy-weighted; those below threshold are terminated. (4) The boundary model iterates on surviving particles until all exit the domain, accumulating scalar flux contributions. Figure \ref{['fig:cfm_mc_2x2']} shows representative learned flow trajectories across a $2\times2$ mesh with colors consistent with the boxes in (a): Purple curves trace ODE integral paths from particles born inside a cell at $t=0$ (blue circles) or entering a cell face (red squares) and orange diamonds mark cell-to-cell transitions.
• Figure 2: Representative conditional sampling behaviors. Figure \ref{['fig:slab_flow_matching']} visualizes conditional flow integration in a homogeneous two-dimensional slab using the learned velocity field $v_\theta$. On the left, latent samples $z_i$ are drawn from a standard Gaussian distribution $\mathcal{N}(0, \mathbf{I})$. On the right, two distinct entry conditions are shown: $(y_0, \mu_0)$ in green and $(\tilde{y}_0, \tilde{\mu}_0)$ in purple, representing particles entering the cell at different positions and angles. For each condition, integrating $v_\theta$ from $t=0$ to $t=1$ transforms the latent samples into valid exit states. The smooth solid curves represent flow trajectories in the learned probability space; the dashed jagged lines depict a potential physical Monte Carlo random walk that the model bypasses. The marginal exit probability distributions along each boundary illustrate that both representations yield statistically equivalent ensemble behavior. Figure \ref{['fig:hexbin_mu_vs_logtime']} compares the joint distribution of exit angle $u_{\mathrm{exit}}$ (x-direction cosine) versus $\log_{10}(\Delta s)$ (logarithmic path length) for standard MC (red density) and GMC (blue density), each generated with 80,000 samples. The hexbin density plot with marginals on the top and right axes demonstrates that GMC captures the complex correlations between exit angle and path length inherent in the transport physics.
• Figure 3: Performance of GMC transport solver on two benchmark problems. Top row: Lattice problem with internal source showing (a) geometric layout on a $7\times7$ cm domain discretized into $112\times112$ cells, (b) scalar flux distribution with MC reference ($10^6$ particles, left) and GMC ($10^6$ particles, right), (c) horizontal and vertical lineout comparisons through the source center. Bottom row: Linearized hohlraum problem with boundary source showing (d) geometric layout with heterogeneous materials on a $1.3\times1.3$ cm domain discretized into $112\times112$ mesh, (e) scalar flux distribution (MC reference with $10^6$ particles, top; GMC with $10^6$ particles, bottom), (f) horizontal and vertical lineout comparisons. The scalar flux is computed via the track-length estimator normalized per source particle.
• Figure 4: Statistical convergence and computational performance of GMC-based transport. Figure \ref{['fig:std_convergence']} shows the convergence of the cell-averaged standard deviation $\bar{\sigma}_\phi$ in scalar flux estimates as a function of sample size $N$, computed from $K=5$ independent runs at each particle count. Both GMC and standard MC exhibit the characteristic $N^{-1/2}$ convergence rate, confirming that the surrogate preserves the statistical properties of Monte Carlo estimation. Figure \ref{['fig:perf_comparison']} demonstrates the algorithmic advantage of the surrogate in a pure scattering medium. Standard MC cost (red) increases linearly with the cell's optical thickness $\tilde{W} = W\sigma_s$ (due to the increased number of scattering events required to escape), the GMC surrogate (blue) maintains constant $O(1)$ computational cost per cell transmission via fixed-step ODE integration.