A Neural Network Based Method to Solve Boundary Value Problems
This work addresses computational challenges in numerical analysis for researchers and engineers, but it appears incremental as it applies existing neural network techniques to a known problem without major breakthroughs.
The paper tackled solving boundary value problems like Laplace and Poisson equations using a neural network-based numerical method, achieving validation through numerical results that demonstrated the method's ability to handle unstructured data points without meshing issues.
A Neural Network (NN) based numerical method is formulated and implemented for solving Boundary Value Problems (BVPs) and numerical results are presented to validate this method by solving Laplace equation with Dirichlet boundary condition and Poisson's equation with mixed boundary conditions. The principal advantage of NN based numerical method is the discrete data points where the field is computed, can be unstructured and do not suffer from issues of meshing like traditional numerical methods such as Finite Difference Time Domain or Finite Element Method. Numerical investigations are carried out for both uniform and non-uniform training grid distributions to understand the efficacy and limitations of this method and to provide qualitative understanding of various parameters involved.