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 Learning (FBOAL) method, which decomposes the domain into sub-domains, for training collocation points based on local maxima and local minima of the PDEs residuals. The effectiveness of FBOAL is demonstrated for non-parameterized and parameterized problems. The comparison with other adaptive sampling methods is also illustrated. The numerical results demonstrate important gains in terms of the accuracy and computational cost of PINNs with FBOAL over the classical PINNs with non-adaptive collocation points. We also apply FBOAL in a complex industrial application involving coupling between mechanical and thermal fields. We show that FBOAL is able to identify the high-gradient locations and even give better predictions for some physical fields than the classical PINNs with collocation points sampled on a pre-adapted finite element mesh built thanks to numerical expert knowledge. From the present study, it is expected that the use of FBOAL will help to improve the conventional numerical solver in the construction of the mesh.

翻译：物理信息神经网络（PINNs）凭借其将物理定律融入模型的能力，在工程各个领域引起了广泛关注。PINNs通过在一组配置点上最小化偏微分方程（PDEs）残差来整合物理约束。这些配置点的分布对PINNs的性能具有巨大影响，而针对这些点的采样方法评估仍是一个活跃的研究课题。本文提出了一种固定预算在线自适应学习方法（FBOAL），该方法将计算域分解为子域，基于PDEs残差的局部极大值和极小值来训练配置点。我们针对非参数化和参数化问题验证了FBOAL的有效性，并展示了与其他自适应采样方法的对比结果。数值结果表明，采用FBOAL的PINNs在计算精度和成本上相较于采用非自适应配置点的经典PINNs有显著提升。我们还将FBOAL应用于涉及机械场与热场耦合的复杂工业案例。研究表明，FBOAL能够识别高梯度区域，甚至在某些物理场上的预测效果优于基于数值专家知识构建的预自适应有限元网格上采样的经典PINNs。预期本研究将有助于改进传统数值求解器在网格构建中的应用。