# Learning Linear Complementarity Systems

arXiv: https://arxiv.org/abs/2112.13284 · HTML: https://mlsanity.com/paper/2112.13284/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 41 references that are in the map.
- First posted on arXiv 25 Dec 2021.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Learning Particle Dynamics for Manipulating Rigid Bodies, Deformable Objects, and Fluids](https://mlsanity.com/paper/1810.01566/) (2018)
- [ContactNets: Learning Discontinuous Contact Dynamics with Smooth, Implicit Representations](https://mlsanity.com/paper/2009.11193/) (2020)
- [Learning Models as Functionals of Signed-Distance Fields for Manipulation Planning](https://mlsanity.com/paper/2110.00792/) (2021)
- [Safe Pontryagin Differentiable Programming](https://mlsanity.com/paper/2105.14937/) (2021)
- [ADD: Analytically Differentiable Dynamics for Multi-Body Systems with Frictional Contact](https://mlsanity.com/paper/2007.00987/) (2020)
- [Fundamental Challenges in Deep Learning for Stiff Contact Dynamics](https://mlsanity.com/paper/2103.15406/) (2021)
- [Real-Time Multi-Contact Model Predictive Control via ADMM](https://mlsanity.com/paper/2109.07076/) (2021)
- [Stabilization of Complementarity Systems via Contact-Aware Controllers](https://mlsanity.com/paper/2008.02104/) (2020)
- [Generalization Bounded Implicit Learning of Nearly Discontinuous Functions](https://mlsanity.com/paper/2112.06881/) (2021)
- [On the complexity of piecewise affine system identification](https://mlsanity.com/paper/1509.02348/) (2015)
- [Piecewise linear regression and classification](https://mlsanity.com/paper/2103.06189/) (2021)

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

- [Smoothed Online Learning for Prediction in Piecewise Affine Systems](https://mlsanity.com/paper/2301.11187/) (2023)
- [CHyLL: Learning Continuous Neural Representations of Hybrid Systems](https://mlsanity.com/paper/2512.10117/) (2025)
- [A neural network-based approach to hybrid systems identification for control](https://mlsanity.com/paper/2404.01814/) (2024)

## Abstract (arXiv)

This paper investigates the learning, or system identification, of a class of piecewise-affine dynamical systems known as linear complementarity systems (LCSs). We propose a violation-based loss which enables efficient learning of the LCS parameterization, without prior knowledge of the hybrid mode boundaries, using gradient-based methods. The proposed violation-based loss incorporates both dynamics prediction loss and a novel complementarity - violation loss. We show several properties attained by this loss formulation, including its differentiability, the efficient computation of first- and second-order derivatives, and its relationship to the traditional prediction loss, which strictly enforces complementarity. We apply this violation-based loss formulation to learn LCSs with tens of thousands of (potentially stiff) hybrid modes. The results demonstrate a state-of-the-art ability to identify piecewise-affine dynamics, outperforming methods which must differentiate through non-smooth linear complementarity problems.