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.

翻译：学习两个函数空间之间的映射已引起广泛研究关注。然而，学习偏微分方程的解算子仍是科学计算领域的挑战。傅里叶神经算子（FNO）近期被提出用于学习解算子，并取得了优异性能。本研究提出一种新型的“伪微分积分算子”（PDIO），以分析和推广FNO中的傅里叶积分算子。PDIO受伪微分算子启发，后者是一种由特定符号表征的广义微分算子。我们利用神经网络参数化该符号，并证明基于神经网络的符号属于光滑符号类。随后验证PDIO为有界线性算子，因此其在索伯列夫空间中连续。我们将PDIO与神经算子结合，开发出“伪微分神经算子”（PDNO），用于学习偏微分方程的非线性解算子。通过达西流动和纳维-斯托克斯方程的实验，验证了所提模型的有效性。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。