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Neural Triangular Transport Maps: A New Approach Towards Sampling in Lattice QCD

Andrey Bryutkin, Youssef Marzouk

TL;DR

The paper tackles the challenge of sampling high-dimensional Boltzmann distributions in lattice field theories by introducing sparse triangular transport maps that exploit locality under periodic boundaries. These maps use Monotone Rectified Neural Networks (MRNN) to parameterize invertible, autoregressive components, and enforce approximate sparsity by conditioning on a local past neighborhood, achieving $O(N)$ complexity with a final Metropolis–Hastings correction to ensure asymptotic exactness. The authors analyze how variable orderings influence sparsity and show that physics-informed orderings (e.g., Checkerboard, Max–Min) improve sampling efficiency, while experiments on the 2D $\phi^4$ theory demonstrate competitive performance against HMC and RealNVP, with favorable parallelizability. The work provides a scalable framework for lattice sampling and lays out a clear path toward extensions to gauge theories and fermionic systems, potentially enabling efficient simulations of more complex quantum field theories.

Abstract

Lattice field theories are fundamental testbeds for computational physics; yet, sampling their Boltzmann distributions remains challenging due to multimodality and long-range correlations. While normalizing flows offer a promising alternative, their application to large lattices is often constrained by prohibitive memory requirements and the challenge of maintaining sufficient model expressivity. We propose sparse triangular transport maps that explicitly exploit the conditional independence structure of the lattice graph under periodic boundary conditions using monotone rectified neural networks (MRNN). We introduce a comprehensive framework for triangular transport maps that navigates the fundamental trade-off between \emph{exact sparsity} (respecting marginal conditional independence in the target distribution) and \emph{approximate sparsity} (computational tractability without fill-ins). Restricting each triangular map component to a local past enables site-wise parallel evaluation and linear time complexity in lattice size $N$, while preserving the expressive, invertible structure. Using $φ^4$ in two dimensions as a controlled setting, we analyze how node labelings (orderings) affect the sparsity and performance of triangular maps. We compare against Hybrid Monte Carlo (HMC) and established flow approaches (RealNVP).

Neural Triangular Transport Maps: A New Approach Towards Sampling in Lattice QCD

TL;DR

The paper tackles the challenge of sampling high-dimensional Boltzmann distributions in lattice field theories by introducing sparse triangular transport maps that exploit locality under periodic boundaries. These maps use Monotone Rectified Neural Networks (MRNN) to parameterize invertible, autoregressive components, and enforce approximate sparsity by conditioning on a local past neighborhood, achieving complexity with a final Metropolis–Hastings correction to ensure asymptotic exactness. The authors analyze how variable orderings influence sparsity and show that physics-informed orderings (e.g., Checkerboard, Max–Min) improve sampling efficiency, while experiments on the 2D theory demonstrate competitive performance against HMC and RealNVP, with favorable parallelizability. The work provides a scalable framework for lattice sampling and lays out a clear path toward extensions to gauge theories and fermionic systems, potentially enabling efficient simulations of more complex quantum field theories.

Abstract

Lattice field theories are fundamental testbeds for computational physics; yet, sampling their Boltzmann distributions remains challenging due to multimodality and long-range correlations. While normalizing flows offer a promising alternative, their application to large lattices is often constrained by prohibitive memory requirements and the challenge of maintaining sufficient model expressivity. We propose sparse triangular transport maps that explicitly exploit the conditional independence structure of the lattice graph under periodic boundary conditions using monotone rectified neural networks (MRNN). We introduce a comprehensive framework for triangular transport maps that navigates the fundamental trade-off between \emph{exact sparsity} (respecting marginal conditional independence in the target distribution) and \emph{approximate sparsity} (computational tractability without fill-ins). Restricting each triangular map component to a local past enables site-wise parallel evaluation and linear time complexity in lattice size , while preserving the expressive, invertible structure. Using in two dimensions as a controlled setting, we analyze how node labelings (orderings) affect the sparsity and performance of triangular maps. We compare against Hybrid Monte Carlo (HMC) and established flow approaches (RealNVP).
Paper Structure (27 sections, 23 equations, 6 figures)

This paper contains 27 sections, 23 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Contributions:
  3. Related Work
  4. The $\phi^4$ Lattice Field Theory
  5. Triangular Maps
  6. Why Neural Networks over Polynomials?
  7. From Conditional Independence to Sparse Triangular Maps
  8. Impact of Variable Orderings on Sparsity Patterns
  9. Lexicographic Ordering
  10. Checkerboard Ordering
  11. Max-Min Distance Ordering
  12. Experiments
  13. Impact of Variable Ordering and Neighborhood Size
  14. Architecture Comparison and Scalability
  15. Statistical Error and Physical Observables
  16. ...and 12 more sections

Figures (6)

  • Figure 1: Dependency structures for a triangular map on a $4 \times 4$ lattice. It contrasts the dense, exact map for a lexicographic ordering (including "fill-in") with the enforced sparse maps for lexicographic and checkerboard ordering.
  • Figure 2: Effective Sample Size (ESS) as a function of training epochs for the nine model configurations. Each panel corresponds to a different variable ordering strategy (Lexicographical, Checkerboard, MaxMin). Within each panel, lines represent models trained with cumulatively increasing neighborhood orders (1st, 2nd, and 3rd). Expanding the conditioning set consistently improves sampling efficiency, with the MaxMin ordering achieving the highest overall performance.
  • Figure 3: Performance comparison of various flow-based architectures. Subfigure (a) plots the ESS during training. Subfigure (b) summarizes the parameter scaling based on the number of lattice sites
  • Figure 4: The statistical error of the measured energy and susceptibility as a function of the number of samples, $N$. The plot compares HMC to the transport maps. The solid lines show the statistical error, while the red dashed line represents the theoretical $1/\sqrt{N}$ behavior of an ideal sampler.
  • Figure 5: Comparison of exact (top row) and enforced sparse (bottom row) dependency structures for a triangular map on a $4 \times 4$ lattice under different orderings. The exact maps reveal the non-local fill-in patterns unique to each ordering, with lexicographic ordering creating a distinctly different dense structure from the more symmetric checkerboard ordering. The sparse maps are, by construction, limited to preceding physical neighbors ($N_p(j)$), highlighting the significant reduction in complexity at the cost of approximation.
  • ...and 1 more figures