物理信息神经网络 (PINN) 已被广泛用于解决各种科学计算问题。然而,大量的训练成本限制了一些实时应用的 PINN。尽管已经提出了一些工作来提高 PINN 的训练效率,但很少有人考虑初始化的影响。为此,我们提出了一种基于新爬行动物初始化的物理信息神经网络(NRPINN)。最初的 Reptile 算法是一种基于标记数据的元学习初始化方法。通过将偏微分方程 (PDE) 作为惩罚项添加到损失函数中,可以使用更少的标记数据甚至没有任何标记数据来训练 PINN。受这个想法的启发,我们提出了新的 Reptile 初始化,以从参数化 PDE 中采样更多任务并调整损失的惩罚项。新的 Reptile 初始化可以通过有监督、无监督和半监督学习从相关任务中获取初始化参数。然后,具有初始化参数的 PINN 可以有效地求解 PDE。此外,新的 Reptile 初始化也可用于 PINN 的变体。最后,我们演示并验证了 NRPINN,同时考虑了正向问题,包括求解 Poisson、Burgers 和薛定谔方程,以及估计 PDE 中未知参数的逆问题。实验结果表明,与其他初始化方法的 PINN 相比,NRPINN 训练速度更快,精度更高。此外,新的 Reptile 初始化也可用于 PINN 的变体。最后,我们演示并验证了 NRPINN,同时考虑了正向问题,包括求解 Poisson、Burgers 和薛定谔方程,以及估计 PDE 中未知参数的逆问题。实验结果表明,与其他初始化方法的 PINN 相比,NRPINN 训练速度更快,精度更高。此外,新的 Reptile 初始化也可用于 PINN 的变体。最后,我们演示并验证了 NRPINN,同时考虑了正向问题,包括求解 Poisson、Burgers 和薛定谔方程,以及估计 PDE 中未知参数的逆问题。实验结果表明,与其他初始化方法的 PINN 相比,NRPINN 训练速度更快,精度更高。
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A novel meta-learning initialization method for physics-informed neural networks
Physics-informed neural networks (PINNs) have been widely used to solve various scientific computing problems. However, large training costs limit PINNs for some real-time applications. Although some works have been proposed to improve the training efficiency of PINNs, few consider the influence of initialization. To this end, we propose a New Reptile initialization-based Physics-Informed Neural Network (NRPINN). The original Reptile algorithm is a meta-learning initialization method based on labeled data. PINNs can be trained with less labeled data or even without any labeled data by adding partial differential equations (PDEs) as a penalty term into the loss function. Inspired by this idea, we propose the new Reptile initialization to sample more tasks from the parameterized PDEs and adapt the penalty term of the loss. The new Reptile initialization can acquire initialization parameters from related tasks by supervised, unsupervised, and semi-supervised learning. Then, PINNs with initialization parameters can efficiently solve PDEs. Besides, the new Reptile initialization can also be used for the variants of PINNs. Finally, we demonstrate and verify the NRPINN considering both forward problems, including solving Poisson, Burgers, and Schrödinger equations, as well as inverse problems, where unknown parameters in the PDEs are estimated. Experimental results show that the NRPINN training is much faster and achieves higher accuracy than PINNs with other initialization methods.