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

Multi-scale DeepOnet (Mscale-DeepOnet) for Mitigating Spectral Bias in Learning High Frequency Operators of Oscillatory Functions B. Wang a, Lizuo Liu b, Wei Cai c, a LCSM(MOE), School of Mathematics and Statistics, Hunan Normal University, Changsha, China b Department of Mathematics, Dartmouth College, Hanover, NH, USA c Department of Mathematics, Southern Methodist University, Dallas, TX, USAAbstract 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.1. Introduction The DeepONet has shown its ability to learn not only explicit mathematical operators like integration and fractional derivatives, but also PDE operators [1, 3, 4, 7, 9]. However, like neural networks such as PINN in solving PDEs, neural operators including DeepOnet and FNO demonstrates the behaviour of spectral bias where learning is preferred for low frequency component of the approximation [11, 12, 14].