# Meshless physics-informed deep learning method for three-dimensional solid mechanics

arXiv: https://arxiv.org/abs/2012.01547 · HTML: https://mlsanity.com/paper/2012.01547/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Dec 2020.

## Built on by (most cited first, 27 of 27)

- [An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensions](https://mlsanity.com/paper/2309.07049/) (2023)
- [A deep learning energy method for hyperelasticity and viscoelasticity](https://mlsanity.com/paper/2201.08690/) (2022)
- [Meta-PDE: Learning to Solve PDEs Quickly Without a Mesh](https://mlsanity.com/paper/2211.01604/) (2022)
- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domains](https://mlsanity.com/paper/2302.04954/) (2023)
- [Calibrating constitutive models with full-field data via physics informed neural networks](https://mlsanity.com/paper/2203.16577/) (2022)
- [Spiking neural networks for nonlinear regression](https://mlsanity.com/paper/2210.03515/) (2022)
- [Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applications](https://mlsanity.com/paper/2310.20425/) (2023)
- [Stochastic stiffness identification and response estimation of Timoshenko beams via physics-informed Gaussian processes](https://mlsanity.com/paper/2309.11875/) (2023)
- [FastVPINNs: Tensor-Driven Acceleration of VPINNs for Complex Geometries](https://mlsanity.com/paper/2404.12063/) (2024)
- [Advancing Generalization in PINNs through Latent-Space Representations](https://mlsanity.com/paper/2411.19125/) (2024)
- [A physics-encoded Fourier neural operator approach for surrogate modeling of divergence-free stress fields in solids](https://mlsanity.com/paper/2408.15408/) (2024)
- [Multiphysics Bench: Benchmarking and Investigating Scientific Machine Learning for Multiphysics PDEs](https://mlsanity.com/paper/2505.17575/) (2025)
- [Probabilistic Physics-integrated Neural Differentiable Modeling for Isothermal Chemical Vapor Infiltration Process](https://mlsanity.com/paper/2311.07798/) (2023)
- [Dynamically configured physics-informed neural network in topology optimization applications](https://mlsanity.com/paper/2312.06993/) (2023)
- [Differentiable Neural-Integrated Meshfree Method for Forward and Inverse Modeling of Finite Strain Hyperelasticity](https://mlsanity.com/paper/2407.11183/) (2024)
- [Utilising physics-guided deep learning to overcome data scarcity](https://mlsanity.com/paper/2211.15664/) (2022)
- [Physics-informed Neural Network: The Effect of Reparameterization in Solving Differential Equations](https://mlsanity.com/paper/2301.12118/) (2023)
- [Parsimonious Universal Function Approximator for Elastic and Elasto-Plastic Cavity Expansion Problems](https://mlsanity.com/paper/2407.19074/) (2024)
- [Physics-informed DeepONet with stiffness-based loss functions for structural response prediction](https://mlsanity.com/paper/2409.00994/) (2024)
- [Improving hp-Variational Physics-Informed Neural Networks for Steady-State Convection-Dominated Problems](https://mlsanity.com/paper/2411.09329/) (2024)
- [Finite-PINN: A Physics-Informed Neural Network with Finite Geometric Encoding for Solid Mechanics](https://mlsanity.com/paper/2412.09453/) (2024)
- [Physics-informed Multiple-Input Operators for efficient dynamic response prediction of structures](https://mlsanity.com/paper/2505.07090/) (2025)
- [FEM-Informed Hypergraph Neural Networks for Efficient Elastoplasticity](https://mlsanity.com/paper/2602.07364/) (2026)
- [Solving nonlinear subsonic compressible flow in infinite domain via multi-stage neural networks](https://mlsanity.com/paper/2601.00342/) (2026)
- [Latent PDE mapping for efficient physics-informed learning across geometries with limited data](https://mlsanity.com/paper/2607.22215/) (2026)
- [A variational physics-informed graph neural network for heterogeneous solid mechanics](https://mlsanity.com/paper/2609.10983/) (2026)

## Abstract (arXiv)

Deep learning and the collocation method are merged and used to solve partial differential equations describing structures' deformation. We have considered different types of materials: linear elasticity, hyperelasticity (neo-Hookean) with large deformation, and von Mises plasticity with isotropic and kinematic hardening. The performance of this deep collocation method (DCM) depends on the architecture of the neural network and the corresponding hyperparameters. The presented DCM is meshfree and avoids any spatial discretization, which is usually needed for the finite element method (FEM). We show that the DCM can capture the response qualitatively and quantitatively, without the need for any data generation using other numerical methods such as the FEM. Data generation usually is the main bottleneck in most data-driven models. The deep learning model is trained to learn the model's parameters yielding accurate approximate solutions. Once the model is properly trained, solutions can be obtained almost instantly at any point in the domain, given its spatial coordinates. Therefore, the deep collocation method is potentially a promising standalone technique to solve partial differential equations involved in the deformation of materials and structural systems as well as other physical phenomena.