A Novel ANN-Based Radial Basis Function Collocation Method for Solving Elliptic Boundary Value Problems

Self Cite

This study introduces a deep neural network approach that utilizes radial basis functions (RBFs) to solve forward and inverse problems in the process of diffusion. The input layer incorporates multiquadric (MQ) RBFs, symbolizing the radial distance between the boundary points on the spacetime boundary and the source points positioned outside the spacetime boundary. The output layer is the initial and boundary data given by analytical solutions of the diffusion equation. Utilizing the concept of the spacetime coordinates, the approximations for forward and backward diffusion problems involve assigning initial data on the bottom or top spacetime boundaries, respectively. As the need for discretization of the governing equation is eliminated, our straightforward approach uses only the provided boundary data and MQ RBFs. To validate the proposed method, various diffusion scenarios, including forward, backward, and inverse problems with noise, are examined. Results indicate that the method can achieve high-precision numerical solutions for solving diffusion problems. Notably, only 1/4 of the initial and boundary conditions are known, yet the method still yields precise results.

Section: Data Preparationmentioning

confidence: 99%

Deep Neural Networks with Spacetime RBF for Solving Forward and Inverse Problems in the Diffusion Process

Ku,

Liu,

Chiu

et al. 2024

Self Cite

“…Such networks are simpler than fully connected ones since they contain only two layers and are easier to train. Secondorder gradient learning algorithms have been developed for such networks [13,[18][19][20]. The results of comparing fully connected networks and radial basis function networks [21] when solving PDEs showed the advantage of radial basis function networks in terms of training time.…”

Section: Introductionmentioning

confidence: 99%

Mathematical Modeling on a Physics-Informed Radial Basis Function Network

Stenkin,

Gorbachenko

2024

The article is devoted to approximate methods for solving differential equations. An approach based on neural networks with radial basis functions is presented. Neural network training algorithms adapted to radial basis function networks are proposed, in particular adaptations of the Nesterov and Levenberg-Marquardt algorithms. The effectiveness of the proposed algorithms is demonstrated for solving model problems of function approximation, differential equations, direct and inverse boundary value problems, and modeling processes in piecewise homogeneous media.

A Spacetime RBF-Based DNNs for Solving Unsaturated Flow Problems

Liu,

Ku,

Chen

2024

This study presents a novel approach for modeling unsaturated flow using deep neural networks (DNNs) integrated with spacetime radial basis functions (RBFs). Traditional methods for simulating unsaturated flow often face challenges in computational efficiency and accuracy, particularly when dealing with nonlinear soil properties and complex boundary conditions. Our proposed model emphasizes the capabilities of DNNs in identifying complex patterns and the accuracy of spacetime RBFs in modeling spatiotemporal data. The training data comprise the initial data, boundary data, and radial distances used to construct the spacetime RBFs. The innovation of this approach is that it introduces spacetime RBFs, eliminating the need to discretize the governing equation of unsaturated flow and directly providing the solution of unsaturated flow across the entire time and space domain. Various error evaluation metrics are thoroughly assessed to validate the proposed method. This study examines a case where, despite incomplete initial and boundary data and noise contamination in the available boundary data, the solution of unsaturated flow can still be accurately determined. The model achieves RMSE, MAE, and MRE values of 10−4, 10−3, and 10−4, respectively, demonstrating that the proposed method is robust for solving unsaturated flow in soils, providing insights beyond those obtainable with traditional methods.