LAUSR

In this paper, we focus on accurately calculating the numerical solution of the integral-differential equation for ruin probability in Erlang(2) renewal risk model with arbitrary claim distribution. Because the analytical solutions of the equation do not usually exist, firstly, using machine learning method in modern artificial intelligence, the activation functions in the ELM model are changed to trigonometric function, the initial conditions in the integral-differential equation are added to the ELM linear solver to get the ITELM model, and the steps and feasibility of the algorithm are strictly deduced in theory. As the analytic solution for the integral-differential equation only exists when the claim is subject to exponential distribution, and the numerical solution can be gotten with the pareto distribution. And, since the number of hidden neurons in the ITELM model is uncertain, a good numerical value of hidden neurons can only be determined through a large number of iterative tests and comparisons in the actual calculation. Then, we construct a multi-objective optimization model and algorithm, which can get the optimal number of hidden neurons to obtain the IOTELM model and algorithm. Then, in the above two cases for exponential distribution and pareto distribution, the optimal number of hidden neurons is calculated by IOTELM model and algorithm, and then corresponding ITELM models and algorithms are constructed to calculate the corresponding ruin probability. Compared with the previous numerical experiments, it can be seen that the numerical accuracy is greatly improved, which verified the versatility, feasibility and superiority of the proposed IOTELM model and algorithm.