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

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.

Entropy stable conservative flux form neural networks

We propose an entropy-stable conservative flux form neural network (CFN) that integrates classical numerical conservation laws into a data-driven framework using the entropy-stable, second-order, and non-oscillatory Kurganov-Tadmor (KT) scheme. The proposed entropy-stable CFN uses slope limiting as a denoising mechanism, ensuring accurate predictions in both noisy and sparse observation environments, as well as in both smooth and discontinuous regions. Numerical experiments demonstrate that the entropy-stable CFN achieves both stability and conservation while maintaining accuracy over extended time domains. Furthermore, it successfully predicts shock propagation speeds in long-term simulations, {\it without} oracle knowledge of later-time profiles in the training data.

A Phase Shift Deep Neural Network for High Frequency Wave Equations in Inhomogeneous Media

In this paper, we propose a phase shift deep neural network (PhaseDNN) which provides a wideband convergence in approximating high frequency solutions of wave equations. The PhaseDNN accounts for the fact that many DNN achieves convergence in the low frequency range first, a series of moderately-sized of DNNs are constructed and trained for selected high frequency ranges. With the help of phase shifts in the frequency domain, each DNN will be trained to approximate the target solution's higher frequency content over a specific range. Due to the phase shift, each DNN achieves the speed of convergence as in the low frequency range. As a result, the proposed PhaseDNN is able to convert high frequency learning to low frequency learning, thus allowing a uniform learning to wideband high frequency functions. The PhaseDNN will then be applied to find the solution of high frequency wave equations in inhomogeneous media. Numerical results have demonstrated the capability of PhaseDNN in learning high frequency functions and oscillatory solutions of Helmholtz equations.

PhaseDNN - A Parallel Phase Shift Deep Neural Network for Adaptive Wideband Learning

In this paper, we propose a phase shift deep neural network (PhaseDNN) which provides a wideband convergence in approximating a high dimensional function during its training of the network. The PhaseDNN utilizes the fact that many DNN achieves convergence in the low frequency range first, thus, a series of moderately-sized of DNNs are constructed and trained in parallel for ranges of higher frequencies. With the help of phase shifts in the frequency domain, implemented through a simple phase factor multiplication on the training data, each DNN in the series will be trained to approximate the target function's higher frequency content over a specific range. Due to the phase shift, each DNN achieves the speed of convergence as in the low frequency range. As a result, the proposed PhaseDNN system is able to convert wideband frequency learning to low frequency learning, thus allowing a uniform learning to wideband high dimensional functions with frequency adaptive training. Numerical results have demonstrated the capability of PhaseDNN in learning information of a target function from low to high frequency uniformly.