# A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate

arXiv: https://arxiv.org/abs/2102.02617 · HTML: https://mlsanity.com/paper/2102.02617/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 61 references that are in the map.
- First posted on arXiv 4 Feb 2021.

## Builds on (most cited first, 23 of 23)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
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- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations](https://mlsanity.com/paper/1703.10230/) (2017)
- [On Loss Functions for Deep Neural Networks in Classification](https://mlsanity.com/paper/1702.05659/) (2017)
- [Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations](https://mlsanity.com/paper/1709.05963/) (2017)
- [Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations](https://mlsanity.com/paper/1804.07010/) (2018)
- [Data-driven discovery of PDEs in complex datasets](https://mlsanity.com/paper/1808.10788/) (2018)
- [A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations](https://mlsanity.com/paper/1806.02957/) (2018)
- [On the Selection of Initialization and Activation Function for Deep Neural Networks](https://mlsanity.com/paper/1805.08266/) (2018)
- [Deep Learning for Genomics: A Concise Overview](https://mlsanity.com/paper/1802.00810/) (2018)
- [Physics-Driven Regularization of Deep Neural Networks for Enhanced Engineering Design and Analysis](https://mlsanity.com/paper/1810.05547/) (2018)
- [Deep learning and the Schr"odinger equation](https://mlsanity.com/paper/1702.01361/) (2017)

## Built on by (most cited first, 13 of 13)

- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [Topology Optimization via Machine Learning and Deep Learning: A Review](https://mlsanity.com/paper/2210.10782/) (2022)
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- [NeuralClothSim: Neural Deformation Fields Meet the Thin Shell Theory](https://mlsanity.com/paper/2308.12970/) (2023)
- [Dynamic Mode Decomposition in Adaptive Mesh Refinement and Coarsening Simulations](https://mlsanity.com/paper/2104.14034/) (2021)
- [Coupled and Uncoupled Dynamic Mode Decomposition in Multi-Compartmental Systems with Applications to Epidemiological and Additive Manufacturing Problems](https://mlsanity.com/paper/2110.06375/) (2021)
- [Learning Efficient Surrogate Dynamic Models with Graph Spline Networks](https://mlsanity.com/paper/2310.16397/) (2023)
- [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)
- [Solving 2-D Helmholtz equation in the rectangular, circular, and elliptical domains using neural networks](https://mlsanity.com/paper/2503.20222/) (2025)
- [Optimization-Informed Neural Networks](https://mlsanity.com/paper/2210.02113/) (2022)
- [Neural Network approximation power on homogeneous and heterogeneous reaction-diffusion equations](https://mlsanity.com/paper/2510.14094/) (2025)

## Abstract (arXiv)

In this paper, a deep collocation method (DCM) for thin plate bending problems is proposed. This method takes advantage of computational graphs and backpropagation algorithms involved in deep learning. Besides, the proposed DCM is based on a feedforward deep neural network (DNN) and differs from most previous applications of deep learning for mechanical problems. First, batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries. A loss function is built with the aim that the governing partial differential equations (PDEs) of Kirchhoff plate bending problems, and the boundary/initial conditions are minimised at those collocation points. A combination of optimizers is adopted in the backpropagation process to minimize the loss function so as to obtain the optimal hyperparameters. In Kirchhoff plate bending problems, the C1 continuity requirement poses significant difficulties in traditional mesh-based methods. This can be solved by the proposed DCM, which uses a deep neural network to approximate the continuous transversal deflection, and is proved to be suitable to the bending analysis of Kirchhoff plate of various geometries.