# Repulsive normalizing flow mixtures for adaptive importance sampling: reliability analysis of complex systems

arXiv: https://arxiv.org/abs/2609.21160 · HTML: https://mlsanity.com/paper/2609.21160/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 63 references that are in the map.
- First posted on arXiv 18 Sep 2026.

## Builds on (most cited first, 17 of 17)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [A Survey of Monte Carlo Methods for Parameter Estimation](https://mlsanity.com/paper/2107.11820/) (2021)
- [Polynomial-Chaos-based Kriging](https://mlsanity.com/paper/1502.03939/) (2015)
- [A Flow-Based Generative Model for Rare-Event Simulation](https://mlsanity.com/paper/2305.07863/) (2023)
- [Hamiltonian MCMC methods for estimating rare events probabilities in high-dimensional problems](https://mlsanity.com/paper/2007.00180/) (2020)
- [Large deviation theory-based adaptive importance sampling for rare events in high dimensions](https://mlsanity.com/paper/2209.06278/) (2022)
- [Rare Event Probability Learning by Normalizing Flows](https://mlsanity.com/paper/2310.19167/) (2023)
- [Accurately Predicting Probabilities of Safety-Critical Rare Events for Intelligent Systems](https://mlsanity.com/paper/2403.13869/) (2024)
- [A Direct Importance Sampling-based Framework for Rare Event Uncertainty Quantification in Non-Gaussian Spaces](https://mlsanity.com/paper/2405.14149/) (2024)
- [Rare event modeling with self-regularized normalizing flows: what can we learn from a single failure?](https://mlsanity.com/paper/2502.21110/) (2025)
- [Efficient rare event estimation for multimodal and high-dimensional system reliability via subset adaptive importance sampling](https://mlsanity.com/paper/2508.00210/) (2025)

## Abstract (arXiv)

Accurate rare-event estimation can be computationally expensive. Classical adaptive importance sampling (IS) schemes often rely on restrictive proposal families and can struggle under multiple failure modes. We propose FAMIS, a flow-based multiple importance sampling (MIS) framework that learns a nonuniform mixture of normalizing flow proposals for rare event estimation. The method does not require presampled failure data or prior knowledge of the number, location, or geometry of the failure modes. Instead, it adaptively learns the mixture through sequential evaluations of the limit state function. To guide training toward the failure domain, FAMIS uses a smooth rare-event surrogate and a tempered target sequence. A defensive exploration mixture improves early-stage coverage, a Rao Blackwellized update adapts the mixture weights, and a Jensen-Shannon repulsion term promotes separation and diversity among the base components. The final failure probability is computed with a deterministic-mixture MIS estimator. Numerical experiments demonstrate that FAMIS accurately approximates quasi-optimal IS densities with fewer training samples and model evaluations, providing stable variance reduction across complex reliability problems.