Physics-informed Neural Networks (PINNs) have recently emerged as a principled way to include prior physical knowledge in form of partial differential equations (PDEs) into neural networks. Although PINNs are generally viewed as mesh-free, current approaches still rely on collocation points within a bounded region, even in settings with spatially sparse signals. Furthermore, if the boundaries are not known, the selection of such a region is difficult and often results in a large proportion of collocation points being selected in areas of low relevance. To resolve this severe drawback of current methods, we present a mesh-free and adaptive approach termed particle-density PINN (pdPINN), which is inspired by the microscopic viewpoint of fluid dynamics. The method is based on the Eulerian formulation and, different from classical mesh-free method, does not require the introduction of Lagrangian updates. We propose to sample directly from the distribution over the particle positions, eliminating the need to introduce boundaries while adaptively focusing on the most relevant regions. This is achieved by interpreting a non-negative physical quantity (such as the density or temperature) as an unnormalized probability distribution from which we sample with dynamic Monte Carlo methods. The proposed method leads to higher sample efficiency and improved performance of PINNs. These advantages are demonstrated on various experiments based on the continuity equations, Fokker-Planck equations, and the heat equation.
翻译:物理知情神经网络(PINNs)最近成为将先前物理知识以部分差异方程式的形式纳入神经网络的一种有原则的方法。尽管PINNs一般被视为无网状,但目前的方法仍然依赖于封闭区域内的合用点,即使在空间稀少信号的环境下也是如此。此外,如果边界不为人知,选择这样一个区域是困难的,而且往往导致在低相关性地区选择大量合用点。为了解决当前方法的这一严重缺陷,我们提出了一种无网状和适应性的方法,称为粒密度PINN(pdPINN),这是从流动动态动态动态的微观观点出发的。这种方法基于Eulerian的配方,不同于传统的网状无网状信号。我们建议直接从粒子位置的分布中抽样抽样,消除引入边界的需要,同时适应最相关的区域。这是通过解释一种非内置式的直方方程式(例如机率、温度的比值的比值的比值的比值)来实现。我们提出的平流式方方方方方方方程的比值分配方法,这些比方式的比方度和正态的比正态的比方方程的比正方方值的比的比方值的比方值的比方值是更高。我们所展示的比方方方方方值的比方方方值的比方值的比方值的比方值的比的比的比的比方方值的比法,这些的比的比的比。