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.

翻译：本文引入了两种新型的渐进保持卷积深度算子网络（APCONs），旨在解决多尺度时间依赖的线性输运问题。我们观察到，采用改进MLP的原始物理信息驱动深度算子网络在维持期望的极限宏观行为时可能表现出不稳定性。因此，这需要利用一种渐进保持的损失函数。受扩散方程中热核的启发，我们提出了一种名为卷积深度算子网络的新架构，该网络在每个滤波器层中使用多个局部卷积操作代替全局热核，并结合池化和激活操作。我们的APCON方法的参数数量与网格大小无关，能够捕捉线性输运问题的扩散行为。最后，通过几个数值算例验证了所提方法的有效性。