伪微分神经算子：用于学习偏微分方程解算子的广义 Fourier 神经算子
Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equations
AI总结：
本文提出由神经网络参数化符号的伪微分积分算子 PDIO，并构建 PDNO 以学习 PDE 非线性解算子，在 Darcy 流和 Navier-Stokes 实验中多数优于现有方法。
摘要：
学习两个函数空间之间的映射已经引起了大量研究关注。然而，在科学计算中，学习偏微分方程（PDE）的解算子仍然是一项挑战。Fourier 神经算子（FNO）最近被提出用于学习解算子，并取得了优异性能。在本研究中，我们提出一种新颖的伪微分积分算子（PDIO），用于分析并推广 FNO 中的 Fourier 积分算子。PDIO 的灵感来自伪微分算子，后者是由特定符号刻画的广义微分算子。我们使用神经网络对该符号进行参数化，并证明基于神经网络的符号包含于一个光滑符号类中。随后，我们验证 PDIO 是有界线性算子，因此在 Sobolev 空间中连续。我们将 PDIO 与神经算子相结合，开发了伪微分神经算子（PDNO），并学习 PDE 的非线性解算子。我们利用 Darcy 流和 Navier-Stokes 方程通过实验验证了所提模型的有效性。所得结果表明，所提出的 PDNO 在大多数实验中优于现有神经算子方法。
英文摘要：
Learning the mapping between two function spaces has garnered considerable research attention. However, learning the solution operator of partial differential equations (PDEs) remains a challenge in scientific computing. Fourier neural operator (FNO) was recently proposed to learn solution operators, and it achieved an excellent performance. In this study, we propose a novel \textit{pseudo-differential integral operator} (PDIO) to analyze and generalize the Fourier integral operator in FNO. PDIO is inspired by a pseudo-differential operator, which is a generalized differential operator characterized by a certain symbol. We parameterize this symbol using a neural network and demonstrate that the neural network-based symbol is contained in a smooth symbol class. Subsequently, we verify that the PDIO is a bounded linear operator, and thus is continuous in the Sobolev space. We combine the PDIO with the neural operator to develop a \textit{pseudo-differential neural operator} (PDNO) and learn the nonlinear solution operator of PDEs. We experimentally validate the effectiveness of the proposed model by utilizing Darcy flow and the Navier-Stokes equation. The obtained results indicate that the proposed PDNO outperforms the existing neural operator approaches in most experiments.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
