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.

翻译：偏微分方程能够描述动力系统中的诸多相关现象。在实际应用中，我们通常需要将形式化的偏微分方程模型与（可能含噪声的）观测数据相结合。这在缺失边界条件或初始条件信息、以及需要识别未知模型参数的情形下尤为重要。近年来，物理信息神经网络已成为解决此类问题的热门工具。然而，在高维场景中，物理信息神经网络常因需要在计算域全空间内密集布设配点而面临计算问题。为解决这一难题，我们提出物理信息边界积分网络——一种将原始问题空间降维一维求解偏微分方程的数据驱动方法。物理信息边界积分网络仅在计算域边界处布设配点，即可获得高精度结果，并在多个实际场景中显著优于物理信息神经网络。通过利用线性微分算子基本解的初等性质，我们提出了一种处理反问题中点源项的原则性且简便的方法。我们通过拉普拉斯方程和泊松方程，在人工数据集和地下水流动重建实际应用中，展示了物理信息边界积分网络的卓越性能。