Xiang Zixue, Peng Wei, Yao Wen, Liu Xu, Zhang Xiaoya
College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China.
Defense Innovation Institute, Chinese Academy of Military Science, Beijing 100071, China; Intelligent Game and Decision Laboratory, China.
Neural Netw. 2025 May;185:107166. doi: 10.1016/j.neunet.2025.107166. Epub 2025 Jan 16.
The Physics-informed Neural Network (PINN) has been a popular method for solving partial differential equations (PDEs) due to its flexibility. However, PINN still faces challenges in characterizing spatio-temporal correlations when solving parametric PDEs due to network limitations. To address this issue, we propose a Physics-Informed Neural Implicit Flow (PINIF) framework, which enables a meshless low-rank representation of the parametric spatio-temporal field based on the expressiveness of the Neural Implicit Flow (NIF), enabling a meshless low-rank representation. In particular, the PINIF framework utilizes the Polynomial Chaos Expansion (PCE) method to quantify the uncertainty in the presence of noise, allowing for a more robust representation of the solution. In addition, PINIF introduces a novel transfer learning framework to speed up the inference of parametric PDEs significantly. The performance of PINIF and PINN is compared on various PDEs especially with variable coefficients and Kolmogorov flow. The comparative results indicate that PINIF outperforms PINN in terms of accuracy and efficiency.
基于物理知识的神经网络（PINN）因其灵活性而成为求解偏微分方程（PDE）的一种流行方法。然而，由于网络限制，PINN在求解参数化PDE时，在表征时空相关性方面仍面临挑战。为了解决这个问题，我们提出了一种基于物理知识的神经隐式流（PINIF）框架，该框架基于神经隐式流（NIF）的表现力，实现了参数化时空场的无网格低秩表示。特别是，PINIF框架利用多项式混沌展开（PCE）方法来量化噪声存在时的不确定性，从而实现对解的更稳健表示。此外，PINIF引入了一种新颖的迁移学习框架，以显著加快参数化PDE的推理速度。在各种PDE上，特别是在变系数和柯尔莫哥洛夫流的情况下，对PINIF和PINN的性能进行了比较。比较结果表明，PINIF在准确性和效率方面优于PINN。
