Dual Neural Network Method for Solving Multiple Definite Integrals.

This study, which examines a calculation method on the basis of a dual neural network for solving multiple definite integrals, addresses the problems of inefficiency, inaccuracy, and difficulty in finding solutions. First, the method offers a dual neural network method to construct a primitive function of the integral problem; it can approximate the primitive function of any given integrand with any precision. On this basis, a neural network calculation method that can solve multiple definite integrals whose upper and lower bounds are arbitrarily given is obtained with repeated applications of the dual neural network to construction of the primitive function. Example simulations indicate that compared with traditional methods, the proposed algorithm is more efficient and precise in obtaining solutions for multiple integrals with unknown integrand, except for the finite input-output data points. The advantages of the proposed method include the following: (1) integral multiplicity shows no influence and restriction on the employment of the method; (2) only a finite set of known sample points is required without the need to know the exact analytical expression of the integrand; and (3) high calculation accuracy is obtained for multiple definite integrals whose integrand is expressed by sample data points.