# Physics informed deep learning for computational elastodynamics without labeled data

arXiv: https://arxiv.org/abs/2006.08472 · HTML: https://mlsanity.com/paper/2006.08472/

## Key facts

- Cited by 80 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2020.

## Built on by (most cited first, 30 of 80)

- [Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges](https://mlsanity.com/paper/2103.11251/) (2021)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [Encoding physics to learn reaction-diffusion processes](https://mlsanity.com/paper/2106.04781/) (2021)
- [Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2107.00940/) (2021)
- [NTopo: Mesh-free Topology Optimization using Implicit Neural Representations](https://mlsanity.com/paper/2102.10782/) (2021)
- [Implicit Neural Spatial Representations for Time-dependent PDEs](https://mlsanity.com/paper/2210.00124/) (2022)
- [A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method](https://mlsanity.com/paper/2206.13103/) (2022)
- [NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators](https://mlsanity.com/paper/2208.11866/) (2022)
- [The mixed deep energy method for resolving concentration features in finite strain hyperelasticity](https://mlsanity.com/paper/2104.09623/) (2021)
- [Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training](https://mlsanity.com/paper/2110.03049/) (2021)
- [Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learning](https://mlsanity.com/paper/2201.12354/) (2022)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [Physics-informed Dyna-Style Model-Based Deep Reinforcement Learning for Dynamic Control](https://mlsanity.com/paper/2108.00128/) (2021)
- [A deep learning energy method for hyperelasticity and viscoelasticity](https://mlsanity.com/paper/2201.08690/) (2022)
- [Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networks](https://mlsanity.com/paper/2308.12716/) (2023)
- [PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems](https://mlsanity.com/paper/2410.01337/) (2024)
- [Constitutive model characterization and discovery using physics-informed deep learning](https://mlsanity.com/paper/2203.09789/) (2022)
- [SeismicNet: Physics-informed neural networks for seismic wave modeling in semi-infinite domain](https://mlsanity.com/paper/2210.14044/) (2022)
- [Imperative Learning: A Self-supervised Neuro-Symbolic Learning Framework for Robot Autonomy](https://mlsanity.com/paper/2406.16087/) (2024)
- [On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Models](https://mlsanity.com/paper/2302.12337/) (2023)
- [P2C2Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamics](https://mlsanity.com/paper/2411.00040/) (2024)
- [Physics-constrained Deep Learning for Robust Inverse ECG Modeling](https://mlsanity.com/paper/2107.12780/) (2021)
- [A Physics Informed Neural Network Approach to Solution and Identification of Biharmonic Equations of Elasticity](https://mlsanity.com/paper/2108.07243/) (2021)
- [Calibrating constitutive models with full-field data via physics informed neural networks](https://mlsanity.com/paper/2203.16577/) (2022)
- [A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs](https://mlsanity.com/paper/2210.03526/) (2022)
- [Physics-informed neural networks for modeling rate- and temperature-dependent plasticity](https://mlsanity.com/paper/2201.08363/) (2022)
- [Physics perception in sloshing scenes with guaranteed thermodynamic consistency](https://mlsanity.com/paper/2106.13301/) (2021)
- [CENN: Conservative energy method based on neural networks with subdomains for solving variational problems involving heterogeneous and complex geometries](https://mlsanity.com/paper/2110.01359/) (2021)
- [Physics-informed Deep Super-resolution for Spatiotemporal Data](https://mlsanity.com/paper/2208.01462/) (2022)
- [Physics-Informed Neural Networks for High-Frequency and Multi-Scale Problems using Transfer Learning](https://mlsanity.com/paper/2401.02810/) (2024)

## Abstract (arXiv)

Numerical methods such as finite element have been flourishing in the past decades for modeling solid mechanics problems via solving governing partial differential equations (PDEs). A salient aspect that distinguishes these numerical methods is how they approximate the physical fields of interest. Physics-informed deep learning is a novel approach recently developed for modeling PDE solutions and shows promise to solve computational mechanics problems without using any labeled data. The philosophy behind it is to approximate the quantity of interest (e.g., PDE solution variables) by a deep neural network (DNN) and embed the physical law to regularize the network. To this end, training the network is equivalent to minimization of a well-designed loss function that contains the PDE residuals and initial/boundary conditions (I/BCs). In this paper, we present a physics-informed neural network (PINN) with mixed-variable output to model elastodynamics problems without resort to labeled data, in which the I/BCs are hardly imposed. In particular, both the displacement and stress components are taken as the DNN output, inspired by the hybrid finite element analysis, which largely improves the accuracy and trainability of the network. Since the conventional PINN framework augments all the residual loss components in a "soft" manner with Lagrange multipliers, the weakly imposed I/BCs cannot not be well satisfied especially when complex I/BCs are present. To overcome this issue, a composite scheme of DNNs is established based on multiple single DNNs such that the I/BCs can be satisfied forcibly in a "hard" manner. The propose PINN framework is demonstrated on several numerical elasticity examples with different I/BCs, including both static and dynamic problems as well as wave propagation in truncated domains. Results show the promise of PINN in the context of computational mechanics applications.