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

2025-10-14

AI Summary

To address the challenges of multimodality, long-range correlations, and memory/expressivity bottlenecks of normalizing flow models in Boltzmann sampling for lattice quantum chromodynamics (Lattice QCD), this work proposes an efficient, invertible generative method based on sparse triangular transport maps. By modeling the conditional independence structure of lattice fields via monotonic rectified neural networks (MRNNs), we introduce an exact–approximate sparsity trade-off framework, combined with node ordering and local past constraints, enabling site-level parallelism and linear time complexity while preserving periodic boundary conditions. Our approach overcomes memory limitations and expressivity constraints of conventional flow models on large lattices. Validated on the two-dimensional φ⁴ theory, it achieves significantly higher sampling efficiency than Hamiltonian Monte Carlo (HMC) and RealNVP, and demonstrates excellent scalability to large system sizes.

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Natural Language Processing: Generation

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Large language models for search
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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).

Problem

• Sampling Boltzmann distributions in lattice field theories
• Overcoming memory constraints in normalizing flows
• Exploiting conditional independence with sparse triangular transport maps

Innovation

• Sparse triangular transport maps exploit lattice conditional independence
• Monotone rectified neural networks enable exact sparsity structure
• Site-wise parallel evaluation achieves linear time complexity