# Nested Mixture of Experts: Cooperative and Competitive Learning of Hybrid Dynamical System

arXiv: https://arxiv.org/abs/2011.10605 · HTML: https://mlsanity.com/paper/2011.10605/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in MoE in reinforcement learning and robotics, one of 12 directions in Mixture-of-experts.
- Builds on 14 of its 21 references that are in the map.
- First posted on arXiv 20 Nov 2020.

## Placed in

- Mixture-of-experts / [MoE in reinforcement learning and robotics](https://mlsanity.com/moe/moe-in-reinforcement-learning-and-robotics/)

## Builds on (most cited first, 14 of 14)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Addressing Function Approximation Error in Actor-Critic Methods](https://mlsanity.com/paper/1802.09477/) (2018)
- [DeepMind Control Suite](https://mlsanity.com/paper/1801.00690/) (2018)
- [Deep Reinforcement Learning in a Handful of Trials using Probabilistic Dynamics Models](https://mlsanity.com/paper/1805.12114/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Neural Network Dynamics for Model-Based Deep Reinforcement Learning with Model-Free Fine-Tuning](https://mlsanity.com/paper/1708.02596/) (2017)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Encoding Physical Constraints in Differentiable Newton-Euler Algorithm](https://mlsanity.com/paper/2001.08861/) (2020)
- [Structured Mechanical Models for Robot Learning and Control](https://mlsanity.com/paper/2004.10301/) (2020)
- [Learning-based Model Predictive Control for Safe Exploration and Reinforcement Learning](https://mlsanity.com/paper/1906.12189/) (2019)
- [Control of A High Performance Bipedal Robot using Viscoelastic Liquid Cooled Actuators](https://mlsanity.com/paper/1906.03811/) (2019)
- [Data-Efficient and Safe Learning for Humanoid Locomotion Aided by a Dynamic Balancing Model](https://mlsanity.com/paper/1906.03812/) (2019)

## Built on by (most cited first, 3 of 3)

- [Multi-Agent Collaboration Mechanisms: A Survey of LLMs](https://mlsanity.com/paper/2501.06322/) (2025)
- [Prismatic World Model: Learning Compositional Dynamics for Planning in Hybrid Systems](https://mlsanity.com/paper/2512.08411/) (2025)
- [Sample Efficient Dynamics Learning for Symmetrical Legged Robots:Leveraging Physics Invariance and Geometric Symmetries](https://mlsanity.com/paper/2210.07329/) (2022)

## Abstract (arXiv)

Model-based reinforcement learning (MBRL) algorithms can attain significant sample efficiency but require an appropriate network structure to represent system dynamics. Current approaches include white-box modeling using analytic parameterizations and black-box modeling using deep neural networks. However, both can suffer from a bias-variance trade-off in the learning process, and neither provides a structured method for injecting domain knowledge into the network. As an alternative, gray-box modeling leverages prior knowledge in neural network training but only for simple systems. In this paper, we devise a nested mixture of experts (NMOE) for representing and learning hybrid dynamical systems. An NMOE combines both white-box and black-box models while optimizing bias-variance trade-off. Moreover, an NMOE provides a structured method for incorporating various types of prior knowledge by training the associative experts cooperatively or competitively. The prior knowledge includes information on robots' physical contacts with the environments as well as their kinematic and dynamic properties. In this paper, we demonstrate how to incorporate prior knowledge into our NMOE in various continuous control domains, including hybrid dynamical systems. We also show the effectiveness of our method in terms of data-efficiency, generalization to unseen data, and bias-variance trade-off. Finally, we evaluate our NMOE using an MBRL setup, where the model is integrated with a model-based controller and trained online.