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We studied the use of deep neural networks (DNNs) in the numerical solution of the oscillatory Fredholm integral equation of the second kind. It is known that the solution of the equation exhibits certain oscillatory behaviors due to the oscillation of the kernel. It was pointed out recently that standard DNNs favor low frequency functions, and as a result, they often produce poor approximation for functions containing high frequency components. We addressed this issue in this study. We first developed a numerical method for solving the equation with DNNs as an approximate solution by designing a numerical quadrature that tailors to computing oscillatory integrals involving DNNs. We proved that the error of the DNN approximate solution of the equation is bounded by the training loss and the quadrature error. We then proposed a multigrade deep learning (MGDL) model to overcome the spectral bias issue of neural networks. Numerical experiments demonstrate that the MGDL model is effective in extracting multiscale information of the oscillatory solution and overcoming the spectral bias issue from which a standard DNN model suffers.
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This content will become publicly available on March 22, 2026
Parameter error analysis for the 3D modified leray-alpha model: analytical and numerical approaches
In this study, we perform a parameter error analysis of the 3D modified Leray-α model using both analytical and numerical approaches. First, we prove the global well-posedness and continuous dependence on the initial data for the assimilated system. Furthermore, under sufficient conditions on the physical parameters and norms of the true solution, we demon- strate that the true solution can be recovered from the approximate solution, with the error determined by the discrepancy between the true and approximate parameters. Numerical simulations are provided to validate the convergence criteria.
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- Award ID(s):
- 2316894
- PAR ID:
- 10625331
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Computational and applied mathematics
- ISSN:
- 2747-5808
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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