# Learning Solution Manifolds for Control Problems via Energy Minimization

arXiv: https://arxiv.org/abs/2203.03432 · HTML: https://mlsanity.com/paper/2203.03432/

## Key facts

- Cited by 1 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 13 of its 35 references that are in the map.
- First posted on arXiv 7 Mar 2022.

## Placed in

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

## Builds on (most cited first, 13 of 13)

- [Generative Adversarial Imitation Learning](https://mlsanity.com/paper/1606.03476/) (2016)
- [An Algorithmic Perspective on Imitation Learning](https://mlsanity.com/paper/1811.06711/) (2018)
- [Implicit Behavioral Cloning](https://mlsanity.com/paper/2109.00137/) (2021)
- [Keep Doing What Worked: Behavioral Modelling Priors for Offline Reinforcement Learning](https://mlsanity.com/paper/2002.08396/) (2020)
- [Imitation Learning as f-Divergence Minimization](https://mlsanity.com/paper/1905.12888/) (2019)
- [Policies Modulating Trajectory Generators](https://mlsanity.com/paper/1910.02812/) (2019)
- [PLATO: Policy Learning using Adaptive Trajectory Optimization](https://mlsanity.com/paper/1603.00622/) (2016)
- [ADD: Analytically Differentiable Dynamics for Multi-Body Systems with Frictional Contact](https://mlsanity.com/paper/2007.00987/) (2020)
- [Guided Policy Search as Approximate Mirror Descent](https://mlsanity.com/paper/1607.04614/) (2016)
- [Motion Planning by Learning the Solution Manifold in Trajectory Optimization](https://mlsanity.com/paper/2107.05842/) (2021)
- [On-Policy Robot Imitation Learning from a Converging Supervisor](https://mlsanity.com/paper/1907.03423/) (2019)
- [Comparison of Distal Teacher Learning with Numerical and Analytical Methods to Solve Inverse Kinematics for Rigid-Body Mechanisms](https://mlsanity.com/paper/2003.00225/) (2020)
- [Non-Linear Trajectory Optimization for Large Step-Ups: Application to the Humanoid Robot Atlas](https://mlsanity.com/paper/2004.12083/) (2020)

## Built on by (most cited first, 1 of 1)

- [EBGAN-MDN: An Energy-Based Adversarial Framework for Multi-Modal Behavior Cloning](https://mlsanity.com/paper/2510.07562/) (2025)

## Abstract (arXiv)

A variety of control tasks such as inverse kinematics (IK), trajectory optimization (TO), and model predictive control (MPC) are commonly formulated as energy minimization problems. Numerical solutions to such problems are well-established. However, these are often too slow to be used directly in real-time applications. The alternative is to learn solution manifolds for control problems in an offline stage. Although this distillation process can be trivially formulated as a behavioral cloning (BC) problem in an imitation learning setting, our experiments highlight a number of significant shortcomings arising due to incompatible local minima, interpolation artifacts, and insufficient coverage of the state space. In this paper, we propose an alternative to BC that is efficient and numerically robust. We formulate the learning of solution manifolds as a minimization of the energy terms of a control objective integrated over the space of problems of interest. We minimize this energy integral with a novel method that combines Monte Carlo-inspired adaptive sampling strategies with the derivatives used to solve individual instances of the control task. We evaluate the performance of our formulation on a series of robotic control problems of increasing complexity, and we highlight its benefits through comparisons against traditional methods such as behavioral cloning and Dataset aggregation (Dagger).