Convolution-Based Model-Solving Method for Three-Dimensional, Unsteady, Partial Differential Equations.

Area of Science:

• Computational Mathematics
• Artificial Intelligence
• Numerical Analysis

Background:

• Neural networks are gaining traction for solving complex mathematical problems.
• Partial differential equations (PDEs) are fundamental in science and engineering.
• Existing numerical methods for PDEs can be computationally intensive.

Purpose of the Study:

• To propose a novel neural network architecture, 3D-PDE-Net, for solving three-dimensional PDEs.
• To develop a robust method capable of handling both linear and nonlinear unsteady PDEs.
• To demonstrate accurate solutions with reduced computational resources.

Main Methods:

• Mathematical derivation of a 3D convolution kernel approximating differential operators.
• Development of the 3D-PDE-Net architecture based on the derived kernel.
• Optimization using the L-BFGS algorithm to minimize normalized mean square error (NMSE).

Main Results:

• 3D-PDE-Net effectively solves three-dimensional PDEs.
• The network achieves high accuracy with a limited number of training samples.
• Demonstrated success in solving both linear and nonlinear unsteady PDEs.

Conclusions:

• 3D-PDE-Net offers an accurate and efficient approach for solving challenging PDEs.
• The method shows significant potential for applications in various scientific and engineering fields.
• This work advances the use of neural networks in computational science.