The paper presents a method for solving initial and boundary value problems using artificial neural networks (ANNs). The approach involves expressing the trial solution of the differential equation as a sum of two parts: one part satisfies the initial/boundary conditions and contains no adjustable parameters, while the other part, involving a feedforward neural network, is trained to satisfy the differential equation. This method is applicable to both ordinary differential equations (ODEs) and partial differential equations (PDEs). The paper discusses the general formulation of the method, including the computation of the gradient of the error function, and provides numerical examples to illustrate its effectiveness. The method is compared with the finite element method for several cases of PDEs, showing that it provides accurate solutions with better interpolation properties. The authors conclude that the method's success is due to the excellent function approximation capabilities of neural networks and the construction of a trial solution that satisfies the boundary conditions. Future research directions include optimizing neural architecture and improving grid sampling techniques.The paper presents a method for solving initial and boundary value problems using artificial neural networks (ANNs). The approach involves expressing the trial solution of the differential equation as a sum of two parts: one part satisfies the initial/boundary conditions and contains no adjustable parameters, while the other part, involving a feedforward neural network, is trained to satisfy the differential equation. This method is applicable to both ordinary differential equations (ODEs) and partial differential equations (PDEs). The paper discusses the general formulation of the method, including the computation of the gradient of the error function, and provides numerical examples to illustrate its effectiveness. The method is compared with the finite element method for several cases of PDEs, showing that it provides accurate solutions with better interpolation properties. The authors conclude that the method's success is due to the excellent function approximation capabilities of neural networks and the construction of a trial solution that satisfies the boundary conditions. Future research directions include optimizing neural architecture and improving grid sampling techniques.