Solutions to elliptic and parabolic problems via finite difference based unsupervised small linear convolutional neural networks

• DOI: 10.1016/j.camwa.2024.08.013
• Publication Date: 2024-08-27T00:27:43Z
• Authors: Adrian Celaya; Keegan Kirk; David Fuentes; Beatrice Riviere
• Keywords: FOS: Computer and information sciences; Computer Science - Machine Learning; Computer Vision and Pattern Recognition (cs.CV); Computer Science - Computer Vision and Pattern Recognition; FOS: Mathematics; Mathematics - Numerical Analysis; Numerical Analysis (math.NA); Machine Learning (cs.LG)

Submitted to CMA, under review<br/>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.<br/>