# Detecting and Mitigating Mode-Collapse for Flow-based Sampling of Lattice Field Theories

## TL;DR

This paper investigates mode-collapse in normalizing flows used for lattice field theory sampling, revealing that it shifts the tunneling problem to training and proposing metrics and strategies to mitigate bias caused by mode-collapse.

## Contribution

It identifies mode-collapse as a training issue in normalizing flows for lattice field theories and introduces metrics and mitigation methods to reduce bias.

## Key findings

- Mode-collapse occurs during training, not sampling.
- A metric to quantify mode-collapse is proposed.
- Mitigation strategies improve estimation of thermodynamic observables.

## Abstract

We study the consequences of mode-collapse of normalizing flows in the context of lattice field theory. Normalizing flows allow for independent sampling. For this reason, it is hoped that they can avoid the tunneling problem of local-update MCMC algorithms for multi-modal distributions. In this work, we first point out that the tunneling problem is also present for normalizing flows but is shifted from the sampling to the training phase of the algorithm. Specifically, normalizing flows often suffer from mode-collapse for which the training process assigns vanishingly low probability mass to relevant modes of the physical distribution. This may result in a significant bias when the flow is used as a sampler in a Markov-Chain or with Importance Sampling. We propose a metric to quantify the degree of mode-collapse and derive a bound on the resulting bias. Furthermore, we propose various mitigation strategies in particular in the context of estimating thermodynamic observables, such as the free energy.