# Differentiable Implicit Soft-Body Physics

arXiv: https://arxiv.org/abs/2102.05791 · HTML: https://mlsanity.com/paper/2102.05791/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 29 references that are in the map.
- First posted on arXiv 11 Feb 2021.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [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)

## Built on by (most cited first, 9 of 9)

- [PAC-NeRF: Physics Augmented Continuum Neural Radiance Fields for Geometry-Agnostic System Identification](https://mlsanity.com/paper/2303.05512/) (2023)
- [PhysTwin: Physics-Informed Reconstruction and Simulation of Deformable Objects from Videos](https://mlsanity.com/paper/2503.17973/) (2025)
- [Gradient-based Bi-level Optimization for Deep Learning: A Survey](https://mlsanity.com/paper/2207.11719/) (2022)
- [Stabilizing Reinforcement Learning in Differentiable Multiphysics Simulation](https://mlsanity.com/paper/2412.12089/) (2024)
- [Vid2Sim: Generalizable, Video-based Reconstruction of Appearance, Geometry and Physics for Mesh-free Simulation](https://mlsanity.com/paper/2506.06440/) (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)
- [Seeing the Wind from a Falling Leaf](https://mlsanity.com/paper/2512.00762/) (2025)
- [Learning Physics-Guided Residual Dynamics for Deformable Object Simulation](https://mlsanity.com/paper/2607.13451/) (2026)
- [Amortising Trajectory Optimisation for Residual MPC via Implicit Contact Differentiation](https://mlsanity.com/paper/2607.24959/) (2026)

## Abstract (arXiv)

We present a differentiable soft-body physics simulator that can be composed with neural networks as a differentiable layer. In contrast to other differentiable physics approaches that use explicit forward models to define state transitions, we focus on implicit state transitions defined via function minimization. Implicit state transitions appear in implicit numerical integration methods, which offer the benefits of large time steps and excellent numerical stability, but require a special treatment to achieve differentiability due to the absence of an explicit differentiable forward pass. In contrast to other implicit differentiation approaches that require explicit formulas for the force function and the force Jacobian matrix, we present an energy-based approach that allows us to compute these derivatives automatically and in a matrix-free fashion via reverse-mode automatic differentiation. This allows for more flexibility and productivity when defining physical models and is particularly important in the context of neural network training, which often relies on reverse-mode automatic differentiation (backpropagation). We demonstrate the effectiveness of our differentiable simulator in policy optimization for locomotion tasks and show that it achieves better sample efficiency than model-free reinforcement learning.