Approximate solution to weighted residual system using Monte Carlo in relation to Differential Equations

Abstract In this research paper we will implement Monte Carlo method for approximate solution to weighted residual system that comes from Galerkin method with regards to differential equations. The problem can be generalized to any system which consists of parametric definite integrals over a particular domain. Galerkin system is reformulated in Monte Carlo notation, enabling efficient solution and comparison with established methods for ordinary differential equations including finite difference approximation and Physics-Informed Neural Networks (PINNs). In order to further illustrate the scope of this approach, we extend it to any system comprising parametric single definite integrals, which can be defined over desired domain.