用小规模线性卷积网络学椭圆方程的间断有限元解，精度高且参数少。
Learning Discontinuous Galerkin Solutions to Elliptic Problems via Small Linear Convolutional Neural Networks
- 用小型线性卷积网络直接学习间断有限元解，避免自动微分和采样点依赖
- 参数量远低于同类方法，精度接近真实解与传统间断有限元解
- 既支持有监督也支持无监督训练，适合追求高效与可解释性的科学计算
近年来，深度学习在求解偏微分方程（PDE）方面受到越来越多关注。然而，许多基于神经网络的方法（如物理信息神经网络）依赖自动微分和配点采样，导致可解释性差、精度较低。为此，本文提出两种使用小型线性卷积神经网络学习椭圆问题间断有限元解的新方法。第一种为有监督方法，依赖标注数据；第二种为无监督方法，无需任何训练数据。两种方法均显著减少参数量，同时在精度上与真实解及标准间断有限元解相当。
原文摘要 · Abstract (English)
In recent years, there has been an increasing interest in using deep learning and neural networks to tackle scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods, such as physics-informed neural networks, depend on automatic differentiation and the sampling of collocation points, which can result in a lack of interpretability and lower accuracy compared to traditional numerical methods. To address this issue, we propose two approaches for learning discontinuous Galerkin solutions to PDEs using small linear convolutional neural networks. Our first approach is supervised and depends on labeled data, while our second approach is unsupervised and does not rely on any training data. In both cases, our methods use substantially fewer parameters than similar numerics-based neural networks while also demonstrating comparable accuracy to the true and DG solutions for elliptic problems.
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