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.

翻译：我们提出$φ-$DeepONet，这是一种物理信息神经算子，旨在学习可能包含间断或呈现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，该定理假设算子及其作用的函数均为连续。然而，许多科学与工程问题涉及天然间断的输入场，以及由材料界面导致的输出场中的强间断与弱间断。在$φ$-DeepONet中，输入中的间断通过多个分支网络处理，而输出中的间断则通过界面的非线性潜在嵌入来学习。该嵌入基于域分解的独热表示构建，并与空间坐标在改进的干路网络中结合。随后，分支网络与干路网络的输出通过点积结合以生成最终解，该解使用物理与界面信息损失函数进行训练。我们在多个一维和二维基准问题上评估$φ$-DeepONet，结果表明即使在存在强界面驱动间断的情况下，它也能提供准确且稳定的预测。