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MscaleFNO: Multi-scale Fourier Neural Operator Learning for Oscillatory Function Spaces

Zhilin You, Zhenli Xu, Wei Cai

TL;DR

The paper addresses learning operators that map oscillatory inputs to oscillatory outputs in high-frequency regimes and proposes a multi-scale Fourier Neural Operator (MscaleFNO) that uses N parallel FNO branches operating at scaled inputs $c_i x$ and $c_i a(x)$, with outputs combined via trainable weights. This architecture extends the multi-scale DNN idea to operator learning and, through sine activations and scale parameters, achieves superior capture of high-frequency content compared to a standard FNO. Numerical experiments on 1D mappings such as $u=\sin(m a)$ and on Helmholtz scattering mappings demonstrate substantial accuracy gains, improved spectral reconstruction, and better generalization to unseen inputs. The results suggest that MscaleFNO can effectively handle high-frequency wave phenomena and holds promise for higher-dimensional Helmholtz problems and inverse scattering applications.

Abstract

In this paper, a multi-scale Fourier neural operator (MscaleFNO) is proposed to reduce the spectral bias of the FNO in learning the mapping between highly oscillatory functions, with application to the nonlinear mapping between the coefficient of the Helmholtz equation and its solution. The MscaleFNO consists of a series of parallel normal FNOs with scaled input of the function and the spatial variable, and their outputs are shown to be able to capture various high-frequency components of the mapping's image. Numerical methods demonstrate the substantial improvement of the MscaleFNO for the problem of wave scattering in the high-frequency regime over the normal FNO with a similar number of network parameters.

MscaleFNO: Multi-scale Fourier Neural Operator Learning for Oscillatory Function Spaces

TL;DR

The paper addresses learning operators that map oscillatory inputs to oscillatory outputs in high-frequency regimes and proposes a multi-scale Fourier Neural Operator (MscaleFNO) that uses N parallel FNO branches operating at scaled inputs and , with outputs combined via trainable weights. This architecture extends the multi-scale DNN idea to operator learning and, through sine activations and scale parameters, achieves superior capture of high-frequency content compared to a standard FNO. Numerical experiments on 1D mappings such as and on Helmholtz scattering mappings demonstrate substantial accuracy gains, improved spectral reconstruction, and better generalization to unseen inputs. The results suggest that MscaleFNO can effectively handle high-frequency wave phenomena and holds promise for higher-dimensional Helmholtz problems and inverse scattering applications.

Abstract

In this paper, a multi-scale Fourier neural operator (MscaleFNO) is proposed to reduce the spectral bias of the FNO in learning the mapping between highly oscillatory functions, with application to the nonlinear mapping between the coefficient of the Helmholtz equation and its solution. The MscaleFNO consists of a series of parallel normal FNOs with scaled input of the function and the spatial variable, and their outputs are shown to be able to capture various high-frequency components of the mapping's image. Numerical methods demonstrate the substantial improvement of the MscaleFNO for the problem of wave scattering in the high-frequency regime over the normal FNO with a similar number of network parameters.
Paper Structure (12 sections, 52 equations, 20 figures)

This paper contains 12 sections, 52 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on Fourier Neural Operator
  3. Theory and Architecture
  4. FNO Network Parameters
  5. Multi-scale FNO
  6. MscaleDNN
  7. Multi-scale Fourier Neural Operator (MscaleFNO)
  8. Mapping between conductivity and solution in Helmholtz equation
  9. Numerical Results
  10. Learning a nonlinear mapping between function $a(x)$ and $u(x)$ - $u=\sin m a$.
  11. Mapping between scatterer properties and solution for Helmholtz Equation
  12. Conclusion

Figures (20)

  • Figure 1: The FNO architecture
  • Figure 2: The MscaleFNO architecture
  • Figure 3: The profile of the input function $a(x)$ and its DFT
  • Figure 4: Error curves of different models during the training process
  • Figure 5: Predicted solution by the normal FNO (left) and MscaleFNO (right) with zoomed-in inset for $x\in[-0.18,-0.12]$
  • ...and 15 more figures

Theorems & Definitions (5)

  • Example 4.1
  • Example 4.2
  • Example 4.3
  • Example 4.4
  • Example 4.5