  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, many neural
network-based methods like PINNs rely on auto differentiation and sampling
collocation points, leading to a lack of interpretability and lower accuracy
than traditional numerical methods. As a result, we propose a fully
unsupervised approach, requiring no training data, to estimate finite
difference solutions for PDEs directly via small linear convolutional neural
networks. Our proposed approach uses substantially fewer parameters than
similar finite difference-based approaches while also demonstrating comparable
accuracy to the true solution for several selected elliptic and parabolic
problems compared to the finite difference method.
