# DiffPD: Differentiable Projective Dynamics

arXiv: https://arxiv.org/abs/2101.05917 · HTML: https://mlsanity.com/paper/2101.05917/

## Key facts

- Cited by 62 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 59 references that are in the map.
- First posted on arXiv 15 Jan 2021.

## Builds on (most cited first, 10 of 10)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [DiffTaichi: Differentiable Programming for Physical Simulation](https://mlsanity.com/paper/1910.00935/) (2019)
- [Learning Particle Dynamics for Manipulating Rigid Bodies, Deformable Objects, and Fluids](https://mlsanity.com/paper/1810.01566/) (2018)
- [Learning to Control PDEs with Differentiable Physics](https://mlsanity.com/paper/2001.07457/) (2020)
- [A Differentiable Physics Engine for Deep Learning in Robotics](https://mlsanity.com/paper/1611.01652/) (2016)
- [ChainQueen: A Real-Time Differentiable Physical Simulator for Soft Robotics](https://mlsanity.com/paper/1810.01054/) (2018)
- [Scalable Differentiable Physics for Learning and Control](https://mlsanity.com/paper/2007.02168/) (2020)
- [SPNets: Differentiable Fluid Dynamics for Deep Neural Networks](https://mlsanity.com/paper/1806.06094/) (2018)

## Built on by (most cited first, 30 of 62)

- [Accelerated Policy Learning with Parallel Differentiable Simulation](https://mlsanity.com/paper/2204.07137/) (2022)
- [Reconstruction and Simulation of Elastic Objects with Spring-Mass 3D Gaussians](https://mlsanity.com/paper/2403.09434/) (2024)
- [LLM and Simulation as Bilevel Optimizers: A New Paradigm to Advance Physical Scientific Discovery](https://mlsanity.com/paper/2405.09783/) (2024)
- [DiffCloth: Differentiable Cloth Simulation with Dry Frictional Contact](https://mlsanity.com/paper/2106.05306/) (2021)
- [Learning Neural Constitutive Laws From Motion Observations for Generalizable PDE Dynamics](https://mlsanity.com/paper/2304.14369/) (2023)
- [GIC: Gaussian-Informed Continuum for Physical Property Identification and Simulation](https://mlsanity.com/paper/2406.14927/) (2024)
- [DiffSkill: Skill Abstraction from Differentiable Physics for Deformable Object Manipulations with Tools](https://mlsanity.com/paper/2203.17275/) (2022)
- [DiffAvatar: Simulation-Ready Garment Optimization with Differentiable Simulation](https://mlsanity.com/paper/2311.12194/) (2023)
- [DiffuseBot: Breeding Soft Robots With Physics-Augmented Generative Diffusion Models](https://mlsanity.com/paper/2311.17053/) (2023)
- [RISP: Rendering-Invariant State Predictor with Differentiable Simulation and Rendering for Cross-Domain Parameter Estimation](https://mlsanity.com/paper/2205.05678/) (2022)
- [Particle-Grid Neural Dynamics for Learning Deformable Object Models from RGB-D Videos](https://mlsanity.com/paper/2506.15680/) (2025)
- [DexDeform: Dexterous Deformable Object Manipulation with Human Demonstrations and Differentiable Physics](https://mlsanity.com/paper/2304.03223/) (2023)
- [ClothesNet: An Information-Rich 3D Garment Model Repository with Simulated Clothes Environment](https://mlsanity.com/paper/2308.09987/) (2023)
- [SAM-RL: Sensing-Aware Model-Based Reinforcement Learning via Differentiable Physics-Based Simulation and Rendering](https://mlsanity.com/paper/2210.15185/) (2022)
- [Adaptive Horizon Actor-Critic for Policy Learning in Contact-Rich Differentiable Simulation](https://mlsanity.com/paper/2405.17784/) (2024)
- [Stabilizing Reinforcement Learning in Differentiable Multiphysics Simulation](https://mlsanity.com/paper/2412.12089/) (2024)
- [Dress-1-to-3: Single Image to Simulation-Ready 3D Outfit with Diffusion Prior and Differentiable Physics](https://mlsanity.com/paper/2502.03449/) (2025)
- [Vid2Sim: Generalizable, Video-based Reconstruction of Appearance, Geometry and Physics for Mesh-free Simulation](https://mlsanity.com/paper/2506.06440/) (2025)
- [UniPhy: Learning a Unified Constitutive Model for Inverse Physics Simulation](https://mlsanity.com/paper/2505.16971/) (2025)
- [VFXMaster: Unlocking Dynamic Visual Effect Generation via In-Context Learning](https://mlsanity.com/paper/2510.25772/) (2025)
- [NeuralFluid: Neural Fluidic System Design and Control with Differentiable Simulation](https://mlsanity.com/paper/2405.14903/) (2024)
- [SoftMAC: Differentiable Soft Body Simulation with Forecast-based Contact Model and Two-way Coupling with Articulated Rigid Bodies and Clothes](https://mlsanity.com/paper/2312.03297/) (2023)
- [MASIV: Toward Material-Agnostic System Identification from Videos](https://mlsanity.com/paper/2508.01112/) (2025)
- [Differentiable Physics Simulations with Contacts: Do They Have Correct Gradients w.r.t. Position, Velocity and Control?](https://mlsanity.com/paper/2207.05060/) (2022)
- [Jade: A Differentiable Physics Engine for Articulated Rigid Bodies with Intersection-Free Frictional Contact](https://mlsanity.com/paper/2309.04710/) (2023)
- [Differentiable Visual Computing for Inverse Problems and Machine Learning](https://mlsanity.com/paper/2312.04574/) (2023)
- [Seeing the Wind from a Falling Leaf](https://mlsanity.com/paper/2512.00762/) (2025)
- [DiffVL: Scaling Up Soft Body Manipulation using Vision-Language Driven Differentiable Physics](https://mlsanity.com/paper/2312.06408/) (2023)
- [MOSIV: Multi-Object System Identification from Videos](https://mlsanity.com/paper/2603.06022/) (2026)
- [Improving Gradient Computation for Differentiable Physics Simulation with Contacts](https://mlsanity.com/paper/2305.00092/) (2023)

## Abstract (arXiv)

We present a novel, fast differentiable simulator for soft-body learning and control applications. Existing differentiable soft-body simulators can be classified into two categories based on their time integration methods: Simulators using explicit time-stepping schemes require tiny time steps to avoid numerical instabilities in gradient computation, and simulators using implicit time integration typically compute gradients by employing the adjoint method and solving the expensive linearized dynamics. Inspired by Projective Dynamics (PD), we present Differentiable Projective Dynamics (DiffPD), an efficient differentiable soft-body simulator based on PD with implicit time integration. The key idea in DiffPD is to speed up backpropagation by exploiting the prefactorized Cholesky decomposition in forward PD simulation. In terms of contact handling, DiffPD supports two types of contacts: a penalty-based model describing contact and friction forces and a complementarity-based model enforcing non-penetration conditions and static friction. We evaluate the performance of DiffPD and observe it is 4-19 times faster compared with the standard Newton's method in various applications including system identification, inverse design problems, trajectory optimization, and closed-loop control. We also apply DiffPD in a reality-to-simulation (real-to-sim) example with contact and collisions and show its capability of reconstructing a digital twin of real-world scenes.