Solutions to Elliptic and Parabolic Problems via Finite Difference Based Unsupervised Small Linear Convolutional Neural Networks

TL;DR

This work addresses solving elliptic and parabolic PDEs by learning finite‑difference–style solutions without any training data, using small linear CNNs that mimic finite difference stencils. A CNN is trained in a fully unsupervised fashion to solve discretized PDEs by minimizing a loss that encodes the five‑point stencil via a kernel $K_\Delta$ and, for parabolic problems, backward Euler in time, with Dirichlet boundaries enforced through a weighted term with $\alpha = h^2/4$. Key contributions include extending to non‑constant diffusion via a dual‑grid construction, time‑dependent problems with per‑step optimization, and demonstrating comparable accuracy to classical finite differences with significantly fewer parameters and linear activation functions. The method provides a transparent, mesh‑free alternative that can serve as a fast initializer or preconditioner for FD solvers, with potential extensions to more PDE types.

Abstract

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.

Figures (8)

• Figure 1: Properties of classical PINNs, numerics informed neural networks (NINNs), and numerical PDEs.
• Figure 2: Sketch of U-Net architecture for different network depths: $0\leq d\leq 4$.
• Figure 3: (Top) Bubble function. (Middle) "Peak" function. (Bottom) Exponential trigonometric function. From left to right, contour plots of true solution, predicted solution, and absolute difference. All predictions and solutions are on a 128$\times$128 grid. Note that $\mu = 10^{-6}$ in the colorbar for the bubble function difference.
• Figure 4: Loss values at each optimization step for the bubble function. Here, we use a network depth of three, a grid size of 128$\times$128, and 2,000 optimization steps.
• Figure 5: From left to right, contour plots of true solution, predicted solution, and absolute difference for the non-constant diffusion problem.