用深度网络+多项式结合，高效高精度求解偏微分方程。
DeePoly: A High-Order Accuracy Scientific Machine Learning Framework for Function Approximation and Solving PDEs
- 先用DNN捕捉全局复杂特征，再用线性优化融合多项式校正。
- 在多种问题上实现高阶精度与计算效率的双重提升。
- 适合需要高精度且无需网格的科学计算场景。
近年来，机器学习方法在科学计算中日益受到关注，尤其在求解偏微分方程（PDEs）方面。然而，基于深度神经网络（DNNs）的方法相较于传统数值格式，在收敛性保证和计算效率上常显不足。本文提出DeePoly框架，将求解范式从纯非凸参数优化转变为两阶段策略：首先使用DNN捕捉复杂全局特征，随后通过线性空间优化，结合DNN提取的特征（Spotter）与多项式基函数（Sniper）。该策略融合了两类方法的优势——DNN擅长逼近复杂全局特征（如高梯度区域），稳定多项式逼近；而多项式基则提供具有收敛保证的高精度局部修正。理论分析与数值实验表明，该方法在多种问题类型中显著提升了高阶精度与计算效率，同时保持无网格、无格式特性。本文亦为开源项目DeePoly提供理论阐述。
原文摘要 · Abstract (English)
Recently, machine learning methods have gained significant traction in scientific computing, particularly for solving Partial Differential Equations (PDEs). However, methods based on deep neural networks (DNNs) often lack convergence guarantees and computational efficiency compared to traditional numerical schemes. This work introduces DeePoly, a novel framework that transforms the solution paradigm from pure non-convex parameter optimization to a two-stage approach: first employing a DNN to capture complex global features, followed by linear space optimization with combined DNN-extracted features (Spotter) and polynomial basis functions (Sniper). This strategic combination leverages the complementary strengths of both methods -- DNNs excel at approximating complex global features (i.e., high-gradient features) and stabilize the polynomial approximation while polynomial bases provide high-precision local corrections with convergence guarantees. Theoretical analysis and numerical experiments demonstrate that this approach significantly enhances both high-order accuracy and efficiency across diverse problem types while maintaining mesh-free and scheme-free properties. This paper also serves as a theoretical exposition for the open-source project DeePoly.
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