提出GEN架构，用函数级建模提升PDE求解的鲁棒性与泛化能力
General Explicit Network (GEN): A novel deep learning architecture for solving partial differential equations
- 以基函数构建函数级解表示，实现点到函数的PDE求解
- 实验显示方案具有高鲁棒性和强泛化能力
- 适合需要稳定解的工程场景和复杂PDE问题
机器学习，尤其是物理信息神经网络（PINNs）及其变体，在求解偏微分方程（PDEs）问题中得到广泛应用。然而，这些方法在学术研究之外的应用仍受限。例如，传统PINN主要采用离散点对点拟合，忽略了真实解可能具备的潜在性质；且使用连续激活函数导致局部特征匹配方程解，但扩展性和鲁棒性较差。本文提出一种通用显式网络（GEN），实现点到函数的PDE求解。该方法通过先验知识选取对应基函数构建“函数”组件，用于拟合解。实验表明，该方法可获得具有高鲁棒性和强扩展性的解。
原文摘要 · Abstract (English)
Machine learning, especially physics-informed neural networks (PINNs) and their neural network variants, has been widely used to solve problems involving partial differential equations (PDEs). The successful deployment of such methods beyond academic research remains limited. For example, PINN methods primarily consider discrete point-to-point fitting and fail to account for the potential properties of real solutions. The adoption of continuous activation functions in these approaches leads to local characteristics that align with the equation solutions while resulting in poor extensibility and robustness. A general explicit network (GEN) that implements point-to-function PDE solving is proposed in this paper. The "function" component can be constructed based on our prior knowledge of the original PDEs through corresponding basis functions for fitting. The experimental results demonstrate that this approach enables solutions with high robustness and strong extensibility to be obtained.
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