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Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equations

Jin Young Shin, Jae Yong Lee, Hyung Ju Hwang

TL;DR

This work recasts operator learning for PDEs in a pseudo-differential operator framework and introduces a pseudo-differential integral operator (PDIO) whose symbol is learned by neural networks. By ensuring the symbol lies in a toroidal symbol class, the authors prove Sobolev-space continuity and construct a time-dependent variant to approximate time-evolving solution operators. The resulting pseudo-differential neural operator (PDNO) combines PDIOs with a neural operator, achieving superior performance to Fourier neural operators and multiwavelet operators on challenging problems such as Darcy flow and Navier–Stokes, while also highlighting limits on non-smooth problems like Burgers' equation. The approach provides a principled, smooth-symbol pathway for operator learning with potential broad impact in scientific computing and PDE-driven modeling.

Abstract

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.

Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equations

TL;DR

This work recasts operator learning for PDEs in a pseudo-differential operator framework and introduces a pseudo-differential integral operator (PDIO) whose symbol is learned by neural networks. By ensuring the symbol lies in a toroidal symbol class, the authors prove Sobolev-space continuity and construct a time-dependent variant to approximate time-evolving solution operators. The resulting pseudo-differential neural operator (PDNO) combines PDIOs with a neural operator, achieving superior performance to Fourier neural operators and multiwavelet operators on challenging problems such as Darcy flow and Navier–Stokes, while also highlighting limits on non-smooth problems like Burgers' equation. The approach provides a principled, smooth-symbol pathway for operator learning with potential broad impact in scientific computing and PDE-driven modeling.

Abstract

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.
Paper Structure (39 sections, 3 theorems, 33 equations, 13 figures, 6 tables)

This paper contains 39 sections, 3 theorems, 33 equations, 13 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Fourier integral operator and PDO
  3. Neural operator
  4. Fourier integral operator
  5. PDO
  6. Difference between Fourier integral operator and PDO
  7. Proposed integral operator : PDIO
  8. Symbol network and PDIO
  9. PDOs on Lg
  10. Propositions on the symbol network and PDIO
  11. Neural operator with PDIOs
  12. PDNO
  13. Time-dependent PDIO
  14. Experiments
  15. Toy example : 1D heat equation
  16. ...and 24 more sections

Key Result

Theorem 1

ruzhansky2009pseudo(Connection between two symbols) Let $0<\rho\leq1$ and $0\leq\delta\leq1$. A symbol $\tilde{a}\in S^m_{\rho,\delta}(\mathbb{T}^n\times\mathbb{Z}^n)$ is a toroidal symbol if and only if there exists a Euclidean symbol $a\in S^m_{\rho,\delta}(\mathbb{T}^n\times\mathbb{R}^n)$ such th

Figures (13)

  • Figure 1: Comparison of the train and the test relative $L^2$ error by time horizon $t=10, ..., 19$ on the Navier-Stokes equation with viscosity $\nu =10^{-5}$. FNO and MWT are highly overfitted, while PDNO is not. See Section \ref{['exp_nonlinear']} for detailed experimental setups regarding the Navier-Stokes equation.
  • Figure 2: An architecture of a PDIO with symbol networks $a_{\theta_1}^{nn}(x)$ and $a_{\theta_2}^{nn}(\xi)$. Considering that FFT and inverse FFT are used, both the input and output are in the form of uniform mesh. Each value $a_{\theta_1}^{nn}(x)$ and $a_{\theta_2}^{nn}(\xi)$ is obtained from separate neural networks.
  • Figure 3: Visualization of the learned symbol from the time-dependent PDIO $a_{\theta_1}^{nn}(x, t)a_{\theta_2}^{nn}(\xi, t)$ (top) and analytic symbol $a(x, \xi, t) = e^{-4\times0.05\pi^2 \xi^2 t}$ (bottom) of the solution operator of the 1D heat equation. Note that learned $a_{\theta_1}^{nn}(x, t)$ is a constant function according to $x$. i.e. $a_{\theta_1}^{nn}(x, t) = c(t)$ (See Figure \ref{['symbol_constant']}). Therefore, it does not require an $x$-coordinate to plot the learned symbol.
  • Figure 4: Example of a prediction on the Navier-Stokes data with $\nu$=1e-5 showing the prediction $w(x, 19)$ from inputs $[w(x, 0), ..., w(x, 9)]$. Each value on the top of the figure is the relative $L^2$ error between the true $w(x, 19)$ and each prediction. A prediction of FNO is more granular than PDNO. We suspect that this is due to non-smooth symbols of FNO.
  • Figure 5: Examples of the real part of learned symbol $a_{ij}^{nn}(\xi)$ from the Navier-Stokes data with $\nu = 1e-5$. $x$-axis and $y$-axis represent frequency domains. As we used real valued functions, the second coordinate is half the first.
  • ...and 8 more figures

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Proposition 1
  • Remark 1
  • proof
  • Theorem 2
  • Definition 5