$ϕ-$DeepONet：一种捕捉不连续性的神经算子
$ϕ-$DeepONet: A Discontinuity Capturing Neural Operator
AI总结：
本文提出$ϕ-$DeepONet，一种物理信息神经算子，用于学习可能包含不连续或非光滑行为的函数空间映射。通过多分支网络处理输入不连续性，非线性潜在嵌入学习输出不连续性，结合一热编码和空间坐标，通过点积生成最终解，使用物理和界面信息损失函数训练。
摘要：
我们提出了$ϕ-$DeepONet，一种物理信息神经算子，旨在学习可能包含不连续或非光滑行为的函数空间之间的映射。经典神经算子基于通用逼近定理，假设算子和其作用的函数都是连续的。然而，许多科学和工程问题涉及自然不连续的输入场以及由材料界面引起的强和弱不连续性。在$ϕ$-DeepONet中，输入的不连续性通过多个分支网络处理，而输出的不连续性则通过非线性潜在嵌入界面学习。该嵌入是从域分解的一热表示与修改后的主干网络中的空间坐标结合而成。分支网络和主干网络的输出通过点积结合，生成最终解，该解通过具有物理和界面信息的损失函数训练。我们对多个一维和二维基准问题进行了评估，证明该方法在强界面驱动不连续性存在的情况下也能提供准确且稳定的预测。
英文摘要：
We present $ϕ-$DeepONet, a physics-informed neural operator designed to learn mappings between function spaces that may contain discontinuities or exhibit non-smooth behavior. Classical neural operators are based on the universal approximation theorem which assumes that both the operator and the functions it acts on are continuous. However, many scientific and engineering problems involve naturally discontinuous input fields as well as strong and weak discontinuities in the output fields caused by material interfaces. In $ϕ$-DeepONet, discontinuities in the input are handled using multiple branch networks, while discontinuities in the output are learned through a nonlinear latent embedding of the interface. This embedding is constructed from a {\it one-hot} representation of the domain decomposition that is combined with the spatial coordinates in a modified trunk network. The outputs of the branch and trunk networks are then combined through a dot product to produce the final solution, which is trained using a physics- and interface-informed loss function. We evaluate $ϕ$-DeepONet on several one- and two-dimensional benchmark problems and demonstrate that it delivers accurate and stable predictions even in the presence of strong interface-driven discontinuities.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
