论文标题
通过功能价值复制内核希尔伯特空间的操作员学习方法用于微分方程
An Operator Learning Approach via Function-valued Reproducing Kernel Hilbert Space for Differential Equations
论文作者
论文摘要
最近的许多工作通过计算输入和解决方案空间之间的反操作员图来解决偏微分方程家族的解决方案。为此，我们在操作员学习模型中结合了功能值繁殖Hilbert空间。我们使用神经网络来参数化Hilbert-Schmidt积分操作员并提出架构。包括几个典型数据集在内的实验表明，即使使用少量数据，提出的架构在线性和非线性部分微分方程上具有理想的精度。通过学习功能空间之间的映射，提出的方法可以在从低分辨率数据中学习后找到高分辨率输入的解决方案。
Much recent work has addressed the solution of a family of partial differential equations by computing the inverse operator map between the input and solution space. Toward this end, we incorporate function-valued reproducing kernel Hilbert spaces in our operator learning model. We use neural networks to parameterize Hilbert-Schmidt integral operator and propose an architecture. Experiments including several typical datasets show that the proposed architecture has desirable accuracy on linear and nonlinear partial differential equations even with a small amount of data. By learning the mappings between function spaces, the proposed method can find the solution of a high-resolution input after learning from lower-resolution data.
