  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.
