基于龙格库塔法的多输出物理信息神经网络模型

MULTI-OUTPUT PHYSICS-INFORMED NEURAL NETWORKS MODEL BASED ON THE RUNGE-KUTTA METHOD

物理信息神经网络(physics-informed neural networks,PINN)由于嵌入了物理先验知识,可以在少量训练数据的情况下获得自动满足物理约束的代理模型,受到了智能科学计算领域的广泛关注.但是,PINN的离散时间模型(PINN-RK)无法同时近似多个物理量相互耦合的偏微分方程系统,限制了其处理复杂多物理场的能力.为了打破这一限制,文章提出了一种基于龙格库塔法的多输出物理信息神经网络(multi-output physics-informed neural networks based on the Runge-Kutta method,MO-PINN-RK),MO-PINN-RK模型在离散时间模型的基础上采用了并行输出的神经网络结构,通过将神经网络划分为多个子网络,建立了多个神经网络输出层.采用不同输出层近似不同物理量的方式,MO-PINN-RK模型不仅可以同时表征多个物理量,而且还能够实现求解偏微分方程系统的目的.另外,MO-PINN-RK克服了PINN离散时间模型仅适用于一维空间的局限性,将其应用范围扩展到了更为普遍的多维空间.为了验证MO-PINN-RK的有效性,文章对圆柱绕流问题进行了流场预测和参数辨识研究.测试结果表明,与PINN相比,MO-PINN-RK在流场预测问题中的准确性获得了提升,其精度至少提高了2倍,而在参数辨识问题中,MO-PINN-RK的相对误差降低了一个数量级.这凸显了MO-PINN-RK在流体动力学领域的卓越能力,为解决复杂问题提供了更准确、更有效的解决方案.

Physics-informed neural networks(PINN)have attracted considerable attention in the field of intelligent scientific computing primarily due to their capacity to incorporate prior knowledge of physics.This outstanding integration allows PINNs to automatically satisfy physical constraints even with limited or zero labeled data.As a result,the applicability and effectiveness of PINN are vastly developed across numerous domains.However,it is worth noting that the discrete time models of PINN,also known as PINN-RK,face a significant limitation in their ability to approximate multiple physical quantities and solve coupled partial differential equation systems simultaneously.This shortcoming hinders its ability to handle complex multi-physics fields,which is a crucial drawback in various practical scenarios.To overcome this limitation,a multi-output physics-informed neural network based on Runge-Kutta method(MO-PINN-RK)is proposed in this paper.MO-PINN-RK,building upon the success of PINN-RK,incorporates a sophisticated parallel neural network architecture,boasting multiple output layers for enhanced performance and accuracy.By associating each output layer with a sub-network and assigning it with different physical quantities,MOPINN-RK can accurately solve the coupled partial differential equation system and predict multiple physical quantities simultaneously.The MO-PINN-RK proposed in this paper overcomes the limitation of PINN-RK that is only applicable to one dimensional problems extending its applicability to more general multi-dimensional problems.To demonstrate the effectiveness of MO-PINN-RK,it is then applied to the flow field prediction and parameter identification of flow around a cylinder.The outcomes unequivocally reveal that MO-PINN-RK surpasses PINN in terms of flow field prediction precision,achieving an enhancement of no less than 2 times.At the same time,MO-PINN-RK reduces the relative error by an order of magnitude in the context of parameter identification.This highlights the exceptional capabilities of MOPINN-RK in the field of fluid dynamics,offering a more accurate and efficient solution for solving complex problems.