Analysis of three dimensional potential problems in non-homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis

Guo, Hongwei; Zhuang, Xiaoying; Chen, Pengwan; Alajlan, Naif; Rabczuk, Timon

doi:10.48550/arxiv.2010.12060

Cited by 3 publications

(5 citation statements)

References 30 publications

(38 reference statements)

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“…The two optimizers are employed in a serial fashion; the Adam optimizer is initially used, and then the limited-memory BFGS (L-BFGS) optimizer with the Strong Wolfe line search method [36,41] is utilized. We found that using both optimizers regularly stabilizes the optimization procedure; a similar conclusion was sketched in other papers in the literature [25,26]. More details are discussed in the following sections.…”

Section: Dnn Modelsupporting

confidence: 85%

“…along with more sophisticated numerical integration schemes to investigate how different sampling techniques impact the model's performance and explore which ones lead to higher accuracy. Guo et al [26] studied the effect of sampling techniques in the context of the deep collocation method (DCM), and it would be interesting to examine this effect in the context of the DEM.…”

Section: Discussion Conclusion and Future Directionsmentioning

confidence: 99%

“…Recently, a number of researchers have proposed and developed frameworks to solve PDEs using DNNs, with a relatively small number of data points or even without the need for any data, by incorporating physical laws into the loss function being minimized [24,25,26,27,28,29,30,31]. The approach used in these papers sometimes is referred to as the deep collocation method (DCM), as collocation points are sampled from the space of interest.…”

Section: Introductionmentioning

confidence: 99%

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A deep learning energy method for hyperelasticity and viscoelasticity

Abueidda,

Koric,

Al-Rub

et al. 2022

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The potential energy formulation and deep learning are merged to solve partial differential equations governing the deformation in hyperelastic and viscoelastic materials. The presented deep energy method (DEM) is self-contained and meshfree. It can accurately capture the three-dimensional (3D) mechanical response without requiring any time-consuming training data generation by classical numerical methods such as the finite element method. Once the model is appropriately trained, the response can be attained almost instantly at any point in the physical domain, given its spatial coordinates. Therefore, the deep energy method is potentially a promising standalone method for solving partial differential equations describing the mechanical deformation of materials or structural systems and other physical phenomena.

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Section: Dnn Modelsupporting

confidence: 85%

Section: Discussion Conclusion and Future Directionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A deep learning energy method for hyperelasticity and viscoelasticity

Abueidda,

Koric,

Al-Rub

et al. 2022

Preprint

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“…Here, we use two optimizers; we start with the Adam optimizer, and then the limited-memory BFGS (L-BFGS) optimizer with the Strong Wolfe line search 65,73 is used. We found that combining both optimizers generally stabilizes the optimization procedure; similar conclusion was outlined in the work of Guo F I G U R E 3 Illustration of some activation functions et al 74 More details are discussed in the following sections. High-throughput computations are done on iForge, which is an HPC cluster hosted at the National Center for Supercomputing Applications (NCSA).…”

Section: Dnn Modelsupporting

confidence: 79%

Meshless physics‐informed deep learning method for three‐dimensional solid mechanics

Abueidda

Korić

2021

Numerical Meth Engineering

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Deep learning (DL) and the collocation method are merged and used to solve partial differential equations (PDEs) 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 DL 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 DCM is potentially a promising standalone technique to solve PDEs involved in the deformation of materials and structural systems as well as other physical phenomena.

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“…[6,17,38,41]). The use of ANN for the numerical solutions of systems of differential equations goes back to [27,28], and the interest on new applications is in nowadays at the forefront of research interest [2,3,8,13,15,18,29,31,32,35,40]. The main focus of such techniques is on the numerical solution of some very difficult problems: for example high dimensional systems with difficult geometries and non-linear.…”

Section: Introductionmentioning

confidence: 99%

Extreme learning machine collocation for the numerical solution of elliptic PDEs with sharp gradients

Calabrò,

Fabiani,

Siettos

2020

Preprint

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We introduce a new numerical method based on machine learning to approximate the solution of elliptic partial differential equations with collocation using a set of sigmoidal functions. We show that a feedforward neural network with a single hidden layer with sigmoidal functions and fixed, random, internal weights and biases can be used to compute accurately a collocation solution. The choice to fix internal weights and bias leads to the so-called Extreme Learning Machine network. We discuss how to determine the range for both internal weights and biases in order to obtain a good underlining approximating space, and we explore the required number of collocation points. We demonstrate the efficiency of the proposed method with several one-dimensional diffusion-advection-reaction problems that exhibit steep behaviors, such as boundary layers. The boundary conditions are imposed directly as collocation equations. We point out that there is no need of training the network, as the proposed numerical approach results to a linear problem that can be easily solved using least-squares.Numerical results show that the proposed method achieves a good accuracy.Finally, we compare the proposed method with finite differences and point out the significant improvements in terms of computational cost, thus avoiding the time-consuming training phase.

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Analysis of three dimensional potential problems in non-homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis

Cited by 3 publications

References 30 publications

A deep learning energy method for hyperelasticity and viscoelasticity

A deep learning energy method for hyperelasticity and viscoelasticity

Meshless physics‐informed deep learning method for three‐dimensional solid mechanics

Extreme learning machine collocation for the numerical solution of elliptic PDEs with sharp gradients

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