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

2025-04-15

AI Summary

This work addresses the spectral bias problem exhibited by DeepOnet when learning the high-frequency nonlinear operator mapping from coefficients to solutions of the Helmholtz equation. We propose Multi-Scale DeepOnet (MS-DeepOnet), the first architecture to explicitly embed multi-scale structures *both* in the branch net—capturing multi-frequency features of the input coefficient function—and in the trunk net—modeling hierarchical high-frequency responses of the solution operator. This dual multi-scale design jointly enhances representation fidelity for high-oscillation components in both input and output. Leveraging a frequency-aware operator learning mechanism, MS-DeepOnet achieves substantial improvements over standard DeepOnet with matched parameter count on wave scattering tasks: prediction error is reduced by up to 42.6%, generalization accuracy improves, and robustness to unseen frequencies increases markedly. Our core contribution is the construction of the first end-to-end multi-scale DeepOnet framework, establishing a new paradigm for operator learning in high-frequency partial differential equations.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Mitigating spectral bias in learning high-frequency operators
• Improving mapping accuracy for oscillatory functions
• Enhancing wave scattering solutions in high-frequency regimes

Innovation

• Multi-scale DeepOnet for high-frequency operators
• Mitigates spectral bias in oscillatory functions
• Improves wave scattering in high-frequency regime