Physics-Informed Neural Networks (PINNs) are a class of deep learning neural networks that learn the response of a physical system without any simulation data, and only by incorporating the governing partial differential equations (PDEs) in their loss function. While PINNs are successfully used for solving forward and inverse problems, their accuracy decreases significantly for parameterized systems. PINNs also have a soft implementation of boundary conditions resulting in boundary conditions not being exactly imposed everywhere on the boundary. With these challenges at hand, we present first-order physics-informed neural networks (FO-PINNs). These are PINNs that are trained using a first-order formulation of the PDE loss function. We show that, compared to standard PINNs, FO-PINNs offer significantly higher accuracy in solving parameterized systems, and reduce time-per-iteration by removing the extra backpropagations needed to compute the second or higher-order derivatives. Additionally, FO-PINNs can enable exact imposition of boundary conditions using approximate distance functions, which pose challenges when applied on high-order PDEs. Through three examples, we demonstrate the advantages of FO-PINNs over standard PINNs in terms of accuracy and training speedup.

翻译：物理信息神经网络（PINNs）是一类深度学习神经网络，它无需任何仿真数据，仅通过在其损失函数中引入控制偏微分方程（PDEs）即可学习物理系统的响应。尽管PINNs已成功用于求解正问题和逆问题，但其在参数化系统上的精度显著下降。此外，PINNs对边界条件采用软约束实现，导致边界条件无法在边界上精确施加。针对这些挑战，我们提出了一阶物理信息神经网络（FO-PINNs）。这类PINNs采用PDE损失函数的一阶形式进行训练。研究表明，与标准PINNs相比，FO-PINNs在求解参数化系统时具有显著更高的精度，并通过消除计算二阶或高阶导数所需的额外反向传播，减少了每次迭代的时间。此外，FO-PINNs能够利用近似距离函数精确施加边界条件，而这一方法在应用于高阶PDE时存在困难。通过三个算例，我们展示了FO-PINNs在精度和训练加速方面相较于标准PINNs的优势。