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




Venkatachalapathy, Pavithra


Journal Title

Journal ISSN

Volume Title




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.



feed-forward neural network, mean squared loss function, neural network, singular ordinary differential equation



This material is made available for use in research, teaching, and private study, pursuant to U.S. Copyright law. The user assumes full responsibility for any use of the materials, including but not limited to, infringement of copyright and publication rights of reproduced materials. Any materials used should be fully credited with its source. All rights are reserved and retained regardless of current or future development or laws that may apply to fair use standards. Permission for publication of this material, in part or in full, must be secured with the author and/or publisher.