Solutions to Elliptic and Parabolic Problems via Finite Difference Based Unsupervised Small Linear Convolutional Neural Networks
This addresses the problem of interpretability and accuracy in neural network-based PDE solvers for scientific computing, though it appears incremental as it builds on finite difference methods.
The paper tackles solving elliptic and parabolic partial differential equations (PDEs) by proposing an unsupervised small linear convolutional neural network approach that requires no training data, achieving comparable accuracy to finite difference methods with fewer parameters.
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.