Dual Neural Network Method for Solving Multiple Definite Integrals.

Area of Science:

• Computational Mathematics
• Artificial Intelligence
• Numerical Analysis

Background:

• Traditional methods for solving multiple definite integrals often suffer from inefficiency and inaccuracy.
• Calculating integrals with unknown or complex integrands presents significant challenges in computational mathematics.
• Existing numerical techniques may struggle with high-dimensional integrals or require extensive computational resources.

Purpose of the Study:

• To develop an efficient and accurate computational method for solving multiple definite integrals.
• To address the limitations of existing methods concerning integrand complexity and computational cost.
• To propose a novel approach utilizing dual neural networks for integral approximation.

Main Methods:

• A dual neural network (DNN) approach was employed to construct the primitive function of integral problems.
• The DNN was designed to approximate any given integrand with arbitrary precision.
• Repeated application of the DNN enabled the calculation of multiple definite integrals with arbitrarily defined bounds.

Main Results:

• The proposed DNN method demonstrated superior efficiency and precision compared to traditional techniques for multiple integrals.
• The method effectively handles integrands defined by a finite set of sample data points, without requiring analytical expressions.
• Integral multiplicity did not impose limitations on the applicability or performance of the DNN-based calculation method.

Conclusions:

• The dual neural network method offers a robust and accurate solution for computing multiple definite integrals.
• This approach significantly enhances computational efficiency and precision, particularly for integrands known only through data points.
• The method's independence from the integrand's analytical form and its scalability with integral multiplicity mark a significant advancement.