Partial differential equations (PDEs) can describe many relevant phenomena in
dynamical systems. In real-world applications, we commonly need to combine
formal PDE models with (potentially noisy) observations. This is especially
relevant in settings where we lack information about boundary or initial
conditions, or where we need to identify unknown model parameters. In recent
years, Physics-informed neural networks (PINNs) have become a popular tool for
problems of this kind. In high-dimensional settings, however, PINNs often
suffer from computational problems because they usually require dense
collocation points over the entire computational domain. To address this
problem, we present Physics-Informed Boundary Integral Networks (PIBI-Nets) as
a data-driven approach for solving PDEs in one dimension less than the original
problem space. PIBI-Nets only need collocation points at the computational
domain boundary, while still achieving highly accurate results, and in several
practical settings, they clearly outperform PINNs. Exploiting elementary
properties of fundamental solutions of linear differential operators, we
present a principled and simple way to handle point sources in inverse
problems. We demonstrate the excellent performance of PIBI-Nets for the Laplace
and Poisson equations, both on artificial data sets and within a real-world
application concerning the reconstruction of groundwater flows.

使用基于物理的深度神经网络的边界积分网络来解决少一个维度的偏微分方程问题，并在地下水流重建方面展示了出色的性能。