Capturing the Diffusive Behavior of the Multiscale Linear Transport Equations by Asymptotic-Preserving Convolutional DeepONets

TL;DR

This paper introduces novel Asymptotic-Preserving Convolutional Deep Operator Networks (APCONs) that effectively model multiscale linear transport equations, maintaining stability and capturing diffusive behavior across scales.

Contribution

The paper proposes two new APCON architectures with grid-size independent parameters, incorporating local convolutions and asymptotic-preserving loss functions for multiscale transport problems.

Findings

APCONs accurately capture diffusive behavior in numerical examples.

The architectures maintain stability where vanilla physics-informed DeepONets fail.

Parameter count remains independent of grid size, enhancing scalability.

Abstract

In this paper, we introduce two types of novel Asymptotic-Preserving Convolutional Deep Operator Networks (APCONs) designed to address the multiscale time-dependent linear transport problem. We observe that the vanilla physics-informed DeepONets with modified MLP may exhibit instability in maintaining the desired limiting macroscopic behavior. Therefore, this necessitates the utilization of an asymptotic-preserving loss function. Drawing inspiration from the heat kernel in the diffusion equation, we propose a new architecture called Convolutional Deep Operator Networks, which employ multiple local convolution operations instead of a global heat kernel, along with pooling and activation operations in each filter layer. Our APCON methods possess a parameter count that is independent of the grid size and are capable of capturing the diffusive behavior of the linear transport problem. Finally, we validate the effectiveness of our methods through several numerical examples.