This article explores operator learning models that can deduce solutions to partial differential equations (PDEs) on arbitrary domains without requiring retraining. We introduce two innovative models rooted in boundary integral equations (BIEs): the Boundary Integral Type Deep Operator Network (BI-DeepONet) and the Boundary Integral Trigonometric Deep Operator Neural Network (BI-TDONet), which are crafted to address PDEs across diverse domains. Once fully trained, these BIE-based models adeptly predict the solutions of PDEs in any domain without the need for additional training. BI-TDONet notably enhances its performance by employing the singular value decomposition (SVD) of bounded linear operators, allowing for the efficient distribution of input functions across its modules. Furthermore, to tackle the issue of function sampling values that do not effectively capture oscillatory and impulse signal characteristics, trigonometric coefficients are utilized as both inputs and outputs in BI-TDONet. Our numerical experiments robustly support and confirm the efficacy of this theoretical framework.

翻译：本文探索了无需重新训练即可推导任意区域上偏微分方程解的算子学习模型。我们引入了两种基于边界积分方程的创新模型：边界积分型深度算子网络与边界积分三角深度算子神经网络，专为处理多样区域上的偏微分方程而设计。这些基于边界积分方程的模型经过充分训练后，能够熟练预测任意区域上偏微分方程的解，无需额外训练。边界积分三角深度算子神经网络通过利用有界线性算子的奇异值分解，实现了输入函数在其模块间的高效分配，从而显著提升了性能。此外，为解决函数采样值难以有效捕捉振荡与脉冲信号特征的问题，边界积分三角深度算子神经网络采用三角系数同时作为输入与输出。我们的数值实验有力地支持并验证了该理论框架的有效性。