Partial differential equations (PDEs) are widely used to describe 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 this kind of problems. 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 require points at the computational domain boundary, while still achieving highly accurate results. Moreover, PIBI-Nets clearly outperform PINNs in several practical settings. 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 datasets and within a real-world application concerning the reconstruction of groundwater flows.

翻译：偏微分方程（PDEs）被广泛用于描述动力系统中的相关现象。在实际应用中，我们通常需要将形式化的PDE模型与（可能含有噪声的）观测数据相结合。这在缺乏边界或初始条件信息，或需要识别未知模型参数的场景中尤为重要。近年来，物理信息神经网络（PINNs）已成为解决此类问题的流行工具。然而，在高维场景中，PINNs常因需要在整个计算域上密集配置配点而面临计算难题。为解决这一问题，我们提出了物理信息边界积分网络（PIBI-Nets）作为一种数据驱动方法，用于在比原问题空间低一维的维度上求解偏微分方程。PIBI-Nets仅需在计算域边界上布点，同时仍能获得高精度的结果。此外，在多种实际场景中，PIBI-Nets明显优于PINNs。利用线性微分算子基本解的基本性质，我们提出了一种原理清晰且简洁的方法来处理反问题中的点源。我们通过人工数据集和一个涉及地下水流重建的实际应用，展示了PIBI-Nets在拉普拉斯方程和泊松方程求解中的优异性能。