This paper presents a method for solving initial and boundary value problems for ordinary differential equations (ODEs) and partial differential equations (PDEs) using artificial neural networks (ANNs). The method involves constructing a trial solution as the sum of two parts: one that satisfies the initial/boundary conditions without adjustable parameters, and another that is a feedforward neural network to be trained to satisfy the differential equation. The neural network's parameters (weights and biases) are adjusted to minimize an error function, allowing the solution to be expressed in a differentiable, closed analytic form. The method is general and can be applied to ODEs, systems of ODEs, and PDEs. It is compared with the finite element method for several PDE problems, showing that the neural approach provides accurate and differentiable solutions with high generalization capabilities. The method is also efficient for implementation on parallel architectures and can be realized in hardware using neuroprocessors. The paper illustrates the method with various examples, including single ODEs, systems of ODEs, and PDEs, and presents results showing the accuracy of the neural approach compared to the finite element method. The method is found to be effective for solving a wide range of differential equations with high accuracy and efficiency.This paper presents a method for solving initial and boundary value problems for ordinary differential equations (ODEs) and partial differential equations (PDEs) using artificial neural networks (ANNs). The method involves constructing a trial solution as the sum of two parts: one that satisfies the initial/boundary conditions without adjustable parameters, and another that is a feedforward neural network to be trained to satisfy the differential equation. The neural network's parameters (weights and biases) are adjusted to minimize an error function, allowing the solution to be expressed in a differentiable, closed analytic form. The method is general and can be applied to ODEs, systems of ODEs, and PDEs. It is compared with the finite element method for several PDE problems, showing that the neural approach provides accurate and differentiable solutions with high generalization capabilities. The method is also efficient for implementation on parallel architectures and can be realized in hardware using neuroprocessors. The paper illustrates the method with various examples, including single ODEs, systems of ODEs, and PDEs, and presents results showing the accuracy of the neural approach compared to the finite element method. The method is found to be effective for solving a wide range of differential equations with high accuracy and efficiency.