A feed-forward neural network framework for the solutions of linear and nonlinear ordinary differential equations

Ordinary Differential Equations (ODEs) play a key role in describing the physical, chemical, and biological processes. The methods for obtaining the solutions to such differential equations are widely studied topic among scientific community. Certain simplified ODEs are tractable by well known analytical techniques while many other demand sophisticated numerical methods. In this thesis we propose a method for solving ordinary differential equations using a framework of Ar tificial Neural Networks (ANN). The unsupervised type of feed-forward ANN is used to find the approximate numerical solutions to the given ODEs up to the desired accuracy. The mean squared loss function is the sum of two terms: the first term satisfies the differential equation; the second term satisfies the initial or boundary conditions. The total loss function is minimized by using general type of quasi-Newton optimization methods to get the desired network output. The approximation capability of the proposed method is verified for varieties of initial or boundary value problems, including linear, nonlinear, singular second-order ODEs, and a system of cou pled nonlinear ODEs with Dirichlet, Neumann and mixed type boundary conditions. Point-wise comparison of our approximations shows strong agreement with available exact solutions and/or Runge-Kutta based numerical solutions. We remark that our proposed algorithm minimizes the learnable network parameters in a given initial or boundary value problems. We believe that the method developed in this thesis can be applied to approximate the solutions to partial differential equations on complex domains.