# DeepPhysics: a physics aware deep learning framework for real-time simulation

arXiv: https://arxiv.org/abs/2109.09491 · HTML: https://mlsanity.com/paper/2109.09491/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 53 references that are in the map.
- First posted on arXiv 17 Sep 2021.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Deep Fluids: A Generative Network for Parameterized Fluid Simulations](https://mlsanity.com/paper/1806.02071/) (2018)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [Learning-Based Animation of Clothing for Virtual Try-On](https://mlsanity.com/paper/1903.07190/) (2019)
- [Animating Pictures with Eulerian Motion Fields](https://mlsanity.com/paper/2011.15128/) (2020)
- [Simulation of hyperelastic materials in real-time using Deep Learning](https://mlsanity.com/paper/1904.06197/) (2019)
- [Learning Soft Tissue Behavior of Organs for Surgical Navigation with Convolutional Neural Networks](https://mlsanity.com/paper/1904.00722/) (2019)
- [Matrix-free GPU implementation of a preconditioned conjugate gradient solver for anisotropic elliptic PDEs](https://mlsanity.com/paper/1302.7193/) (2013)

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

- [MAgNET: A Graph U-Net Architecture for Mesh-Based Simulations](https://mlsanity.com/paper/2211.00713/) (2022)
- [Thermodynamics-informed neural networks for physically realistic mixed reality](https://mlsanity.com/paper/2210.13414/) (2022)
- [Convolution, aggregation and attention based deep neural networks for accelerating simulations in mechanics](https://mlsanity.com/paper/2212.01386/) (2022)
- [Real-time elastic partial shape matching using a neural network-based adjoint method](https://mlsanity.com/paper/2303.09343/) (2023)
- [Learning Nonlinear Finite Element Solution Operators using Multilayer Perceptrons and Energy Minimization](https://mlsanity.com/paper/2412.04596/) (2024)
- [Data-driven registration and modeling of brain deformation for image-guided neurosurgery](https://mlsanity.com/paper/2602.10155/) (2026)

## Abstract (arXiv)

Real-time simulation of elastic structures is essential in many applications, from computer-guided surgical interventions to interactive design in mechanical engineering. The Finite Element Method is often used as the numerical method of reference for solving the partial differential equations associated with these problems. Yet, deep learning methods have recently shown that they could represent an alternative strategy to solve physics-based problems 1,2,3. In this paper, we propose a solution to simulate hyper-elastic materials using a data-driven approach, where a neural network is trained to learn the non-linear relationship between boundary conditions and the resulting displacement field. We also introduce a method to guarantee the validity of the solution. In total, we present three contributions: an optimized data set generation algorithm based on modal analysis, a physics-informed loss function, and a Hybrid Newton-Raphson algorithm. The method is applied to two benchmarks: a cantilever beam and a propeller. The results show that our network architecture trained with a limited amount of data can predict the displacement field in less than a millisecond. The predictions on various geometries, topologies, mesh resolutions, and boundary conditions are accurate to a few micrometers for non-linear deformations of several centimeters of amplitude.