用随机特征方法高效求解偏微分方程，计算更简单、更快。
Solving Partial Differential Equations with Random Feature Models
- 基于随机特征的浅层模型替代深层网络，降低计算开销。
- 在多个典型PDE测试集上表现优于现有方法，尤其在点数多时优势明显。
- 理论保证强且实现简便，适合需要快速求解的工程应用。
近年来，基于机器学习的偏微分方程（PDE）求解器受到广泛关注。当前进展主要依赖深度神经网络（如物理信息神经网络PINNs）和核方法。本文提出一种基于随机特征的高效求解框架。随机特征法最初用于近似大规模核机，也可视为一种浅层神经网络。我们提供了该方法的误差分析，并在多个典型PDE基准上进行了全面数值实验。相比现有先进求解器在大量采样点下计算困难的问题，本方法显著降低了计算复杂度。此外，其实现简单，无需额外计算资源。凭借理论保障与计算优势，该方法被证明对求解PDE具有高效性。
原文摘要 · Abstract (English)
Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and kernel method. In this paper, we introduce a random feature based framework toward efficiently solving PDEs. Random feature method was originally proposed to approximate large-scale kernel machines and can be viewed as a shallow neural network as well. We provide an error analysis for our proposed method along with comprehensive numerical results on several PDE benchmarks. In contrast to the state-of-the-art solvers that face challenges with a large number of collocation points, our proposed method reduces the computational complexity. Moreover, the implementation of our method is simple and does not require additional computational resources. Due to the theoretical guarantee and advantages in computation, our approach is proven to be efficient for solving PDEs.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。