Learning Only On Boundaries: a Physics-Informed Neural operator for
  Solving Parametric Partial Differential Equations in Complex Geometries
Zhiwei Fang Sifan Wang Paris Perdikaris
Recently deep learning surrogates and neural operators have shown promise in
solving partial differential equations (PDEs). However, they often require a
large amount of training data and are limited to bounded domains. In this work,
we present a novel physics-informed neural operator method to solve
parametrized boundary value problems without labeled data. By reformulating the
PDEs into boundary integral equations (BIEs), we can train the operator network
solely on the boundary of the domain. This approach reduces the number of
required sample points from $O(N^d)$ to $O(N^{d-1})$, where $d$ is the domain's
dimension, leading to a significant acceleration of the training process.
Additionally, our method can handle unbounded problems, which are unattainable
for existing physics-informed neural networks (PINNs) and neural operators. Our
numerical experiments show the effectiveness of parametrized complex geometries
and unbounded problems.