# Causal Navigation by Continuous-time Neural Networks

arXiv: https://arxiv.org/abs/2106.08314 · HTML: https://mlsanity.com/paper/2106.08314/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Placed in Imitation learning methods, one of 13 directions in Offline RL and imitation learning from data.
- Builds on 50 of its 100 references that are in the map.
- First posted on arXiv 15 Jun 2021.

## Placed in

- Offline RL and imitation learning from data / [Imitation learning methods](https://mlsanity.com/offlinerl/imitation-learning-methods/)

## Builds on (most cited first, 30 of 50)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling](https://mlsanity.com/paper/1412.3555/) (2014)
- [Generative Adversarial Imitation Learning](https://mlsanity.com/paper/1606.03476/) (2016)
- [An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling](https://mlsanity.com/paper/1803.01271/) (2018)
- [Learning Quadrupedal Locomotion over Challenging Terrain](https://mlsanity.com/paper/2010.11251/) (2020)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Learning by Cheating](https://mlsanity.com/paper/1912.12294/) (2019)
- [Learning to Explore using Active Neural SLAM](https://mlsanity.com/paper/2004.05155/) (2020)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Causal Confusion in Imitation Learning](https://mlsanity.com/paper/1905.11979/) (2019)
- [Algorithmic Recourse: from Counterfactual Explanations to Interventions](https://mlsanity.com/paper/2002.06278/) (2020)
- [Causality for Machine Learning](https://mlsanity.com/paper/1911.10500/) (2019)
- [One-Shot Imitation Learning](https://mlsanity.com/paper/1703.07326/) (2017)
- [Cognitive Mapping and Planning for Visual Navigation](https://mlsanity.com/paper/1702.03920/) (2017)
- [Semi-parametric Topological Memory for Navigation](https://mlsanity.com/paper/1803.00653/) (2018)
- [Neural Topological SLAM for Visual Navigation](https://mlsanity.com/paper/2005.12256/) (2020)
- [Latent ODEs for Irregularly-Sampled Time Series](https://mlsanity.com/paper/1907.03907/) (2019)
- [Liquid Time-constant Networks](https://mlsanity.com/paper/2006.04439/) (2020)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Neural Jump Stochastic Differential Equations](https://mlsanity.com/paper/1905.10403/) (2019)
- [Can Autonomous Vehicles Identify, Recover From, and Adapt to Distribution Shifts?](https://mlsanity.com/paper/2006.14911/) (2020)
- [Learning to Control PDEs with Differentiable Physics](https://mlsanity.com/paper/2001.07457/) (2020)
- [Learning Exploration Policies for Navigation](https://mlsanity.com/paper/1903.01959/) (2019)
- [Imitation Learning from Imperfect Demonstration](https://mlsanity.com/paper/1901.09387/) (2019)
- [Visual Representations for Semantic Target Driven Navigation](https://mlsanity.com/paper/1805.06066/) (2018)
- [Lipschitz Recurrent Neural Networks](https://mlsanity.com/paper/2006.12070/) (2020)

## Built on by (most cited first, 23 of 23)

- [Liquid Structural State-Space Models](https://mlsanity.com/paper/2209.12951/) (2022)
- [Closed-form Continuous-time Neural Models](https://mlsanity.com/paper/2106.13898/) (2021)
- [Dataset Distillation with Convexified Implicit Gradients](https://mlsanity.com/paper/2302.06755/) (2023)
- [Trajectory Flow Matching with Applications to Clinical Time Series Modeling](https://mlsanity.com/paper/2410.21154/) (2024)
- [Evolution of Neural Tangent Kernels under Benign and Adversarial Training](https://mlsanity.com/paper/2210.12030/) (2022)
- [Gaussian Splatting to Real World Flight Navigation Transfer with Liquid Networks](https://mlsanity.com/paper/2406.15149/) (2024)
- [Sparse Flows: Pruning Continuous-depth Models](https://mlsanity.com/paper/2106.12718/) (2021)
- [GoTube: Scalable Stochastic Verification of Continuous-Depth Models](https://mlsanity.com/paper/2107.08467/) (2021)
- [BarrierNet: A Safety-Guaranteed Layer for Neural Networks](https://mlsanity.com/paper/2111.11277/) (2021)
- [On the Forward Invariance of Neural ODEs](https://mlsanity.com/paper/2210.04763/) (2022)
- [A Survey of Methods, Challenges and Perspectives in Causality](https://mlsanity.com/paper/2302.00293/) (2023)
- [Are All Vision Models Created Equal? A Study of the Open-Loop to Closed-Loop Causality Gap](https://mlsanity.com/paper/2210.04303/) (2022)
- [Learning with Chemical versus Electrical Synapses -- Does it Make a Difference?](https://mlsanity.com/paper/2401.08602/) (2023)
- [Entangled Residual Mappings](https://mlsanity.com/paper/2206.01261/) (2022)
- [Liquid Fourier Latent Dynamics Networks for fast GPU-based numerical simulations in computational cardiology](https://mlsanity.com/paper/2408.09818/) (2024)
- [Towards Cooperative Flight Control Using Visual-Attention](https://mlsanity.com/paper/2212.11084/) (2022)
- [TLNets: Transformation Learning Networks for long-range time-series prediction](https://mlsanity.com/paper/2305.15770/) (2023)
- [Leveraging Low-Rank and Sparse Recurrent Connectivity for Robust Closed-Loop Control](https://mlsanity.com/paper/2310.03915/) (2023)
- [Interpreting Neural Policies with Disentangled Tree Representations](https://mlsanity.com/paper/2210.06650/) (2022)
- [Infrastructure-based End-to-End Learning and Prevention of Driver Failure](https://mlsanity.com/paper/2303.12224/) (2023)
- [Robustness Analysis of Continuous-Depth Models with Lagrangian Techniques](https://mlsanity.com/paper/2308.12192/) (2023)
- [The power of dynamic causality in observer-based design for soft sensor applications](https://mlsanity.com/paper/2509.11336/) (2025)
- [Real-time fall detection based on vision for low-power edge platforms](https://mlsanity.com/paper/2607.12909/) (2026)

## Abstract (arXiv)

Imitation learning enables high-fidelity, vision-based learning of policies within rich, photorealistic environments. However, such techniques often rely on traditional discrete-time neural models and face difficulties in generalizing to domain shifts by failing to account for the causal relationships between the agent and the environment. In this paper, we propose a theoretical and experimental framework for learning causal representations using continuous-time neural networks, specifically over their discrete-time counterparts. We evaluate our method in the context of visual-control learning of drones over a series of complex tasks, ranging from short- and long-term navigation, to chasing static and dynamic objects through photorealistic environments. Our results demonstrate that causal continuous-time deep models can perform robust navigation tasks, where advanced recurrent models fail. These models learn complex causal control representations directly from raw visual inputs and scale to solve a variety of tasks using imitation learning.