# CusADi: A GPU Parallelization Framework for Symbolic Expressions and Optimal Control

arXiv: https://arxiv.org/abs/2408.09662 · HTML: https://mlsanity.com/paper/2408.09662/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in RL training of language and agent models, one of 15 directions in Deep reinforcement learning methods.
- Builds on 20 of its 49 references that are in the map.
- First posted on arXiv 19 Aug 2024.

## Placed in

- Deep reinforcement learning methods / [RL training of language and agent models](https://mlsanity.com/deeprl/rl-training-of-language-and-agent-models/)

## Builds on (most cited first, 20 of 20)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Isaac Gym: High Performance GPU-Based Physics Simulation For Robot Learning](https://mlsanity.com/paper/2108.10470/) (2021)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Learning robust perceptive locomotion for quadrupedal robots in the wild](https://mlsanity.com/paper/2201.08117/) (2022)
- [Residual Policy Learning](https://mlsanity.com/paper/1812.06298/) (2018)
- [Extreme Parkour with Legged Robots](https://mlsanity.com/paper/2309.14341/) (2023)
- [OSQP: An Operator Splitting Solver for Quadratic Programs](https://mlsanity.com/paper/1711.08013/) (2017)
- [Robot Parkour Learning](https://mlsanity.com/paper/2309.05665/) (2023)
- [DTC: Deep Tracking Control](https://mlsanity.com/paper/2309.15462/) (2023)
- [Perceptive Locomotion through Nonlinear Model Predictive Control](https://mlsanity.com/paper/2208.08373/) (2022)
- [Crocoddyl: An Efficient and Versatile Framework for Multi-Contact Optimal Control](https://mlsanity.com/paper/1909.04947/) (2019)
- [Deep Value Model Predictive Control](https://mlsanity.com/paper/1910.03358/) (2019)
- [GPU Acceleration of ADMM for Large-Scale Quadratic Programming](https://mlsanity.com/paper/1912.04263/) (2019)
- [Tailoring Solution Accuracy for Fast Whole-body Model Predictive Control of Legged Robots](https://mlsanity.com/paper/2407.10789/) (2024)
- [CACTO: Continuous Actor-Critic with Trajectory Optimization -- Towards global optimality](https://mlsanity.com/paper/2211.06625/) (2022)
- [Learning to Optimize in Model Predictive Control](https://mlsanity.com/paper/2212.02603/) (2022)
- [Integrating Model-Based Footstep Planning with Model-Free Reinforcement Learning for Dynamic Legged Locomotion](https://mlsanity.com/paper/2408.02662/) (2024)
- [MPCGPU: Real-Time Nonlinear Model Predictive Control through Preconditioned Conjugate Gradient on the GPU](https://mlsanity.com/paper/2309.08079/) (2023)
- [Inherently robust suboptimal MPC for autonomous racing with anytime feasible SQP](https://mlsanity.com/paper/2401.02194/) (2024)

## Built on by (most cited first, 6 of 6)

- [Differentiable Model Predictive Control on the GPU](https://mlsanity.com/paper/2510.06179/) (2025)
- [Safe Large-Scale Robust Nonlinear MPC in Milliseconds via Reachability-Constrained System Level Synthesis on the GPU](https://mlsanity.com/paper/2604.07644/) (2026)
- [Generalizable and Fast Surrogates: Model Predictive Control of Articulated Soft Robots using Physics-Informed Neural Networks](https://mlsanity.com/paper/2502.01916/) (2025)
- [Accelerating and Scaling MPC-Guided Reinforcement Learning for Humanoid Locomotion and Manipulation](https://mlsanity.com/paper/2606.05687/) (2026)
- [TurboMPC: Fast, Scalable, and Differentiable Model Predictive Control on the GPU](https://mlsanity.com/paper/2606.24039/) (2026)
- [RL-Augmented MPC for Non-Gaited Legged and Hybrid Locomotion](https://mlsanity.com/paper/2603.10878/) (2026)

## Abstract (arXiv)

The parallelism afforded by GPUs presents significant advantages in training controllers through reinforcement learning (RL). However, integrating model-based optimization into this process remains challenging due to the complexity of formulating and solving optimization problems across thousands of instances. In this work, we present CusADi, an extension of the CasADi symbolic framework to support the parallelization of arbitrary closed-form expressions on GPUs with CUDA. We also formulate a closed-form approximation for solving general optimal control problems, enabling large-scale parallelization and evaluation of MPC controllers. Our results show a ten-fold speedup relative to similar MPC implementation on the CPU, and we demonstrate the use of CusADi for various applications, including parallel simulation, parameter sweeps, and policy training.