Solutions to Elliptic and Parabolic Problems via Finite Difference Based
  Unsupervised Small Linear Convolutional Neural Networks
Adrian Celaya Keegan Kirk David Fuentes Beatrice Riviere
In recent years, there has been a growing interest in leveraging deep
learning and neural networks to address scientific problems, particularly in
solving partial differential equations (PDEs). However, current neural
network-based PDE solvers often rely on extensive training data or labeled
input-output pairs, making them prone to challenges in generalizing to
out-of-distribution examples. To mitigate the generalization gap encountered by
conventional neural network-based methods in estimating PDE solutions, we
formulate a fully unsupervised approach, requiring no training data, to
estimate finite difference solutions for PDEs directly via small convolutional
neural networks. Our proposed algorithms demonstrate a comparable accuracy to
the true solution for several selected elliptic and parabolic problems compared
to the finite difference method.