RBF Neural Network Approximation Method

In mathematics, there exists a class of integrals where the integrand itself contains integral terms. Several methods can be employed to solve such integrals, one of which is the RBF (Radial Basis Function) Neural Network Approximation Method. This approach involves dividing the integration interval into multiple subintervals and selecting sample points within each subinterval. The function values at these sample points are then computed to form the training dataset. The neural network is trained using these samples to approximate the integrand function across the entire integration domain. From an implementation perspective, this method typically involves defining the RBF network architecture with appropriate activation functions (such as Gaussian or multiquadric functions), determining optimal weights through training algorithms like gradient descent or pseudoinverse methods, and validating the approximation accuracy. The trained network then provides a smooth functional approximation that can be integrated using standard numerical integration techniques like Gaussian quadrature or Simpson's rule. This method finds broad practical applications in solving various types of complex integrals, serving as a powerful tool for both mathematical research and engineering applications where traditional analytical methods prove insufficient. The neural network approximation effectively handles cases where the integrand exhibits complex behavior or lacks closed-form expressions.