# Variational Inference MPC using Normalizing Flows and Out-of-Distribution Projection

arXiv: https://arxiv.org/abs/2205.04667 · HTML: https://mlsanity.com/paper/2205.04667/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 40 references that are in the map.
- First posted on arXiv 10 May 2022.

## Builds on (most cited first, 26 of 26)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Joint 2D-3D-Semantic Data for Indoor Scene Understanding](https://mlsanity.com/paper/1702.01105/) (2017)
- [Variational Lossy Autoencoder](https://mlsanity.com/paper/1611.02731/) (2016)
- [Learning Likelihoods with Conditional Normalizing Flows](https://mlsanity.com/paper/1912.00042/) (2019)
- [Likelihood Regret: An Out-of-Distribution Detection Score For Variational Auto-encoder](https://mlsanity.com/paper/2003.02977/) (2020)
- [Sample-efficient Cross-Entropy Method for Real-time Planning](https://mlsanity.com/paper/2008.06389/) (2020)
- [Information Theoretic Model Predictive Control: Theory and Applications to Autonomous Driving](https://mlsanity.com/paper/1707.02342/) (2017)
- [Learning Sampling Distributions for Robot Motion Planning](https://mlsanity.com/paper/1709.05448/) (2017)
- [Continuous-Time Gaussian Process Motion Planning via Probabilistic Inference](https://mlsanity.com/paper/1707.07383/) (2017)
- [Variational Inference MPC for Bayesian Model-based Reinforcement Learning](https://mlsanity.com/paper/1907.04202/) (2019)
- [Detecting Out-of-Distribution Inputs to Deep Generative Models Using Typicality](https://mlsanity.com/paper/1906.02994/) (2019)
- [Stein Variational Model Predictive Control](https://mlsanity.com/paper/2011.07641/) (2020)
- [Adaptive importance sampling for control and inference](https://mlsanity.com/paper/1505.01874/) (2015)
- [Deeply Informed Neural Sampling for Robot Motion Planning](https://mlsanity.com/paper/1809.10252/) (2018)
- [Learning Implicit Sampling Distributions for Motion Planning](https://mlsanity.com/paper/1806.01968/) (2018)
- [Parallelised Diffeomorphic Sampling-based Motion Planning](https://mlsanity.com/paper/2108.11775/) (2021)
- [Stochastic Optimal Control as Approximate Input Inference](https://mlsanity.com/paper/1910.03003/) (2019)
- [Variational Inference MPC using Tsallis Divergence](https://mlsanity.com/paper/2104.00241/) (2021)
- [Dual Online Stein Variational Inference for Control and Dynamics](https://mlsanity.com/paper/2103.12890/) (2021)
- [Improving Variational Autoencoder based Out-of-Distribution Detection for Embedded Real-time Applications](https://mlsanity.com/paper/2107.11750/) (2021)
- [MPC-MPNet: Model-Predictive Motion Planning Networks for Fast, Near-Optimal Planning under Kinodynamic Constraints](https://mlsanity.com/paper/2101.06798/) (2021)
- [Keep it Simple: Data-efficient Learning for Controlling Complex Systems with Simple Models](https://mlsanity.com/paper/2102.02493/) (2021)

## Built on by (most cited first, 13 of 13)

- [Motion Planning Diffusion: Learning and Adapting Robot Motion Planning with Diffusion Models](https://mlsanity.com/paper/2412.19948/) (2024)
- [Bootstrapped Model Predictive Control](https://mlsanity.com/paper/2503.18871/) (2025)
- [FlowMP: Learning Motion Fields for Robot Planning with Conditional Flow Matching](https://mlsanity.com/paper/2503.06135/) (2025)
- [Learning Sampling Distributions for Model Predictive Control](https://mlsanity.com/paper/2212.02587/) (2022)
- [Graph Reinforcement Learning for Network Control via Bi-Level Optimization](https://mlsanity.com/paper/2305.09129/) (2023)
- [Constrained Stein Variational Trajectory Optimization](https://mlsanity.com/paper/2308.12110/) (2023)
- [Deep Model Predictive Optimization](https://mlsanity.com/paper/2310.04590/) (2023)
- [Sampling for Model Predictive Trajectory Planning in Autonomous Driving using Normalizing Flows](https://mlsanity.com/paper/2404.09657/) (2024)
- [Design of Restricted Normalizing Flow towards Arbitrary Stochastic Policy with Computational Efficiency](https://mlsanity.com/paper/2412.12894/) (2024)
- [Safe Beyond the Horizon: Efficient Sampling-based MPC with Neural Control Barrier Functions](https://mlsanity.com/paper/2502.15006/) (2025)
- [Improving the Resilience of Quadrotors in Underground Environments by Combining Learning-based and Safety Controllers](https://mlsanity.com/paper/2509.02808/) (2025)
- [Constrained Stein Variational Gradient Descent for Robot Perception, Planning, and Identification](https://mlsanity.com/paper/2506.00589/) (2025)
- [Test-time reward-guided alignment of language models by importance sampling on pre-logit space](https://mlsanity.com/paper/2510.26219/) (2025)

## Abstract (arXiv)

We propose a Model Predictive Control (MPC) method for collision-free navigation that uses amortized variational inference to approximate the distribution of optimal control sequences by training a normalizing flow conditioned on the start, goal and environment. This representation allows us to learn a distribution that accounts for both the dynamics of the robot and complex obstacle geometries. We can then sample from this distribution to produce control sequences which are likely to be both goal-directed and collision-free as part of our proposed FlowMPPI sampling-based MPC method. However, when deploying this method, the robot may encounter an out-of-distribution (OOD) environment, i.e. one which is radically different from those used in training. In such cases, the learned flow cannot be trusted to produce low-cost control sequences. To generalize our method to OOD environments we also present an approach that performs projection on the representation of the environment as part of the MPC process. This projection changes the environment representation to be more in-distribution while also optimizing trajectory quality in the true environment. Our simulation results on a 2D double-integrator and a 3D 12DoF underactuated quadrotor suggest that FlowMPPI with projection outperforms state-of-the-art MPC baselines on both in-distribution and OOD environments, including OOD environments generated from real-world data.