论文标题

固定预算的在线自适应学习,用于物理知识的神经网络。迈向参数化问题推理

Fixed-budget online adaptive learning for physics-informed neural networks. Towards parameterized problem inference

论文作者

Nguyen, Thi Nguyen Khoa, Dairay, Thibault, Meunier, Raphaël, Millet, Christophe, Mougeot, Mathilde

论文摘要

物理知识的神经网络(PINNS)由于其将物理定律纳入模型的能力而引起了各个工程领域的关注。 PINN通过最大程度地减少一组搭配点上的偏微分方程(PDE)残差来整合物理约束。这些搭配点的分布似乎对PINN的性能产生了巨大影响,对这些点的采样方法的评估仍然是一个主题。在本文中,我们提出了一种固定预算在线自适应学习(FBOAL)方法,该方法将域分解为子域,以基于局部最大值和PDES残差的本地最小值训练搭配点。对于非参数化和参数化问题,证明了FBOAL的有效性。还说明了与其他自适应抽样方法的比较。数值结果表明,与具有非自适应搭配点的经典PINn相比,PINN的准确性和计算成本的重要性。我们还将FBOAL应用于涉及机械和热场之间耦合的复杂工业应用中。我们表明,与经典的PINN相比,FBOAL能够识别高梯度位置,甚至可以为某些物理领域提供更好的预测,而这些PINNS凭借数字专家知识在预先适应的有限元网格上采样了搭配点。从本研究中可以预期,FBOAL的使用将有助于改善网格构造中的常规数值求解器。

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.

扫码加入交流群

加入微信交流群

微信交流群二维码

扫码加入学术交流群,获取更多资源