A neural network-based PDE solving algorithm with high precision

Jiang, Zichao; Jiang, Junyang; Yao, Qinghe; Yang, Guochen

doi:10.21203/rs.3.rs-2266425/v1

Cited by 3 publications

(4 citation statements)

References 29 publications

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“…The performance improvement of AdamGOA_ShCNN is compared to typical techniques, such as ANN [7], DNN [5], and ResNet [2].…”

Section: Comparative Methodsmentioning

confidence: 99%

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Research on the Data-Driven Differential Equation-Solving Algorithm Based on Artificial Intelligence

Guoxing Si

2024

jes

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Data-driven DEs has gained popularity in the past few years. This work proposes the new framework, named Adam Gannet Optimization Algorithm (AdamGOA), that combines Adam Optimization and Gannet Optimization Algorithm (GOA) to improve a stability, solve higher order Differential Equations (DE) and accuracy of DE. Adam is a first-order gradient-based methods, optimizes stochastic objectives using adaptive lower-order moments. In contrast, GOA represents a different distinct action of a gannets mathematically during foraging and is employed to facilitate exploitation and exploration. In addition, a Shepard Convolutional Neural Network (ShCNN) processed data to construct meta-data and estimate derivatives. After that, the unified integral form is established to determine optimal structure. Heterogeneous parameters are used to estimate and are labeled as constants or variables. Furthermore, the experimental findings showed that the AdamGOA_ ShCNN beat leading models in Accuracy, Convergence, and Mean Square Error (MSE), with values of 0.989, 4, and 0.539, respectively.

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“…The performance improvement of AdamGOA_ShCNN is compared to typical techniques, such as ANN [7], DNN [5], and ResNet [2].…”

Section: Comparative Methodsmentioning

confidence: 99%

“…Zichao Jiang et al [5] Tamirat Temesgen Dufera [7] has introduced Deep Artificial Neural Networks to solve a system of ordinary DE. Increasing neuron size improves accuracy but requires more iterations to learn the parameters.…”

Section: Literature Surveymentioning

confidence: 99%

Research on the Data-Driven Differential Equation-Solving Algorithm Based on Artificial Intelligence

Guoxing Si

2024

jes

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“…In practice, data-driven models are trained on numerical simulation results and approximate a solution to the system of equations. The inference step of the successfully trained model takes a fraction of the computational resources compared to the full mechanistic model (63,64).…”

Section: Neural Network As Surrogate Modelsmentioning

confidence: 99%

A review of mechanistic learning in mathematical oncology

Metzcar,

Jutzeler,

Macklin

et al. 2024

Front. Immunol.

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Mechanistic learning refers to the synergistic combination of mechanistic mathematical modeling and data-driven machine or deep learning. This emerging field finds increasing applications in (mathematical) oncology. This review aims to capture the current state of the field and provides a perspective on how mechanistic learning may progress in the oncology domain. We highlight the synergistic potential of mechanistic learning and point out similarities and differences between purely data-driven and mechanistic approaches concerning model complexity, data requirements, outputs generated, and interpretability of the algorithms and their results. Four categories of mechanistic learning (sequential, parallel, extrinsic, intrinsic) of mechanistic learning are presented with specific examples. We discuss a range of techniques including physics-informed neural networks, surrogate model learning, and digital twins. Example applications address complex problems predominantly from the domain of oncology research such as longitudinal tumor response predictions or time-to-event modeling. As the field of mechanistic learning advances, we aim for this review and proposed categorization framework to foster additional collaboration between the data- and knowledge-driven modeling fields. Further collaboration will help address difficult issues in oncology such as limited data availability, requirements of model transparency, and complex input data which are embraced in a mechanistic learning framework

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“…One of the two primary types of artificial neural networks, a feedforward neural network, is distinguished by the way information is processed and sent between its various layers. Feedforward neural networks are structured as a sequence of interconnected layers (1)(2)(3)(4)(5)(6)(7). The initial layer is connected to the network's input, and each subsequent layer is linked to the one preceding it.…”

Section: Feed-forward Neural Networkmentioning

confidence: 99%

Artificial neural network with the Levenberg-Marquardtalgorithm for numerical solution of two-dimension Poisson’sequation

Thander

2023

BIJSCIT

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This study introduces an Artificial Neural Network (ANN) framework to address the two-dimensional Poisson’sequation within a rectangular domain. It places a focus on the training process of a neural network with threelayers, incorporating hidden neurons. The feedforward ANN is trained using MATLAB, which calculates weights forall neurons within the network structure. These acquired weights are subsequently applied in the trained networkmodel to make predictions for the desired output of a specific partial differential equation. The architecture of theANN consists of three layers: one input layer, one hidden layer, and one output layer. In this study, we specificallyemploy an ANN configuration with 50 hidden neurons. The training process is executed using MATLAB, utilizing theLevenberg–Marquardt algorithm (LMA) for optimization. Furthermore, the study encompasses the development ofsurface and contour plots that illustrate the computational solution of the partial differential equation. Additionally,error functions are graphed to assess the effectiveness of the ANN model.

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A neural network-based PDE solving algorithm with high precision

Abstract: λ A DNN-based algorithm that solves the multi-diagonal linear equations is proposed. λ We employed an iteration method that decreased the error of the numerical solution to 10− 7. λ The computational efficiency of the proposed method is 2 to 10 times of the classic algorithms.

Cited by 3 publications

References 29 publications

Research on the Data-Driven Differential Equation-Solving Algorithm Based on Artificial Intelligence

Research on the Data-Driven Differential Equation-Solving Algorithm Based on Artificial Intelligence

A review of mechanistic learning in mathematical oncology

Artificial neural network with the Levenberg-Marquardtalgorithm for numerical solution of two-dimension Poisson’sequation

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