Physics-informed neural networks (PINNs) represent a significant advancement in scientific machine learning by integrating fundamental physical laws into their architecture through loss functions. PINNs have been successfully applied to solve various forward and inverse problems in partial differential equations (PDEs). However, a notable challenge can emerge during the early training stages when solving inverse problems. Specifically, data losses remain high while PDE residual losses are minimized rapidly, thereby exacerbating the imbalance between loss terms and impeding the overall efficiency of PINNs. To address this challenge, this study proposes a novel framework termed data-guided physics-informed neural networks (DG-PINNs). The DG-PINNs framework is structured into two distinct phases: a pre-training phase and a fine-tuning phase. In the pre-training phase, a loss function with only the data loss is minimized in a neural network. In the fine-tuning phase, a composite loss function, which consists of the data loss, PDE residual loss, and, if available, initial and boundary condition losses, is minimized in the same neural network. Notably, the pre-training phase ensures that the data loss is already at a low value before the fine-tuning phase commences. This approach enables the fine-tuning phase to converge to a minimal composite loss function with fewer iterations compared to existing PINNs. To validate the effectiveness, noise-robustness, and efficiency of DG-PINNs, extensive numerical investigations are conducted on inverse problems related to several classical PDEs, including the heat equation, wave equation, Euler--Bernoulli beam equation, and Navier--Stokes equation. The numerical results demonstrate that DG-PINNs can accurately solve these inverse problems and exhibit robustness against noise in training data.

本研究提出了一种名为数据引导的物理信息神经网络（DG-PINNs）的新框架，通过两个不同的阶段，即预训练阶段和微调阶段，有效地解决神经网络在解决反问题时出现的数据损失高和整体效率低的挑战，并通过对经典偏微分方程的反问题进行广泛的数值验证，证明了DG-PINNs的准确性和对训练数据中噪声的鲁棒性。

数据引导的物理信息神经网络用于解决偏微分方程中的反问题