Physics-Informed Neural Networks (PINNs) have gained much attention in various fields of engineering thanks to their capability of incorporating physical laws into the models. PINNs integrate the physical constraints by minimizing the partial differential equations (PDEs) residuals on a set of collocation points. The distribution of these collocation points appears to have a huge impact on the performance of PINNs and the assessment of the sampling methods for these points is still an active topic. In this paper, we propose a Fixed-Budget Online Adaptive Mesh Learning (FBOAML) method, which decomposes the domain into sub-domains, for training collocation points based on local maxima and local minima of the PDEs residuals. The stopping criterion is based on a data set of reference, which leads to an adaptive number of iterations for each specific problem. The effectiveness of FBOAML is demonstrated in the context of non-parameterized and parameterized problems. The impact of the hyper-parameters in FBOAML is investigated in this work. The comparison with other adaptive sampling methods is also illustrated. The numerical results demonstrate important gains in terms of accuracy of PINNs with FBOAML over the classical PINNs with non-adaptive collocation points. We also apply FBOAML in a complex industrial application involving coupling between mechanical and thermal fields. We show that FBOAML is able to identify the high-gradient location and even give better prediction for some physical fields than the classical PINNs with collocation points taken on a pre-adapted finite element mesh.

翻译：物理进化神经网络(PINNs)由于能够将物理法纳入模型,在各个工程领域都得到了很大关注。 PINNs通过将部分差异方程式(PDEs)残余物在一组合用点上最小化,整合了物理限制。这些合用点的分布似乎对PINNs的性能和对这些点取样方法的评估产生了巨大影响。在本文中,我们提出了一个固定预算在线调适精度网学习(FBOAML)方法,该方法甚至将域分解为子域,用于根据PDEs残余物的当地最大值和当地微型值来培训合用点。停止标准以一组参考数据为基础,这导致每个具体问题的迭代数的调整。FBOAM的有效性表现在非量化和参数前参数化问题中。FBOAM的超常度度度度度度值(FBOAM)在这项工作中,与其他适应性能性能的精确度比,还展示了FLIM的精确度。