Multi-scale DeepOnet (Mscale-DeepOnet) for Mitigating Spectral Bias in Learning High Frequency Operators of Oscillatory Functions

TL;DR

The paper tackles spectral bias in neural operators like DeepOnet when learning high-frequency mappings such as the nonlinear relation between the dielectric coefficient $a(x)$ and the scattering field $u(x)$ in Helmholtz problems. It proposes Mscale-DeepOnet by embedding a multi-scale DNN into both the branch and trunk nets and extends to complex-valued learning to capture multiple spectral components of the Helmholtz operator, using trainable scales. Across nonlinear mapping and Helmholtz scattering tasks, it demonstrates substantial improvements over standard DeepOnet with comparable parameter counts, particularly in high-frequency regimes, thanks to frequency-scale decomposition and scale-parameterization. This approach offers a more accurate and robust neural-operator framework for high-frequency oscillatory PDE mappings, with potential impact on wave scattering and related operator-learning applications.

Abstract

In this paper, a multi-scale DeepOnet (Mscale-DeepOnet) is proposed to reduce the spectral bias of the DeepOnet in learning high-frequency mapping between highly oscillatory functions, with an application to the nonlinear mapping between the coefficient of the Helmholtz equation and its solution. The Mscale-DeepOnet introduces the multiscale neural network in the branch and trunk networks of the original DeepOnet, the resulting Mscale-DeepOnet is shown to be able to capture various high-frequency components of the mapping itself and its image. Numerical results demonstrate the substantial improvement of the Mscale-DeepOnet for the problem of wave scattering in the high-frequency regime over the normal DeepOnet with a similar number of network parameters.

Figures (12)

• Figure 1: Approximations of the nonlinear mapping $\mathcal{G}_{50}[a]$ on a training data.
• Figure 2: Approximations of the nonlinear mapping $\mathcal{G}_{50}[a]$ on a testing data.
• Figure 3: Loss on training (left) and testing (right) data for the various cases.
• Figure 4: Training results using DeepONet ($S_{\rm branch}=1, S_{\rm trunk}=1$) for $M=10, k=100$.
• Figure 5: Training results using multi-scale DeepONet ($S_{\rm branch}=1, S_{\rm trunk}=10$) for the case $M=10, k=100$.