This work addresses the development of a physics-informed neural network (PINN) with a loss term derived from a discretized time-dependent reduced-order system. In this work, first, the governing equations are discretized using a finite difference scheme (whereas, any other discretization technique can be adopted), then projected on a reduced or latent space using the Proper Orthogonal Decomposition (POD)-Galerkin approach and next, the residual arising from discretized reduced order equation is considered as an additional loss penalty term alongside the data-driven loss term using different variants of deep learning method such as Artificial neural network (ANN), Long Short-Term Memory based neural network (LSTM). The LSTM neural network has been proven to be very effective for time-dependent problems in a purely data-driven environment. The current work demonstrates the LSTM network's potential over ANN networks in physics-informed neural networks (PINN) as well. The potential of using discretized governing equations instead of continuous form lies in the flexibility of input to the PINN. Different sizes of data ranging from small, medium to big datasets are used to assess the potential of discretized-physics-informed neural networks when there is very sparse or no data available. The proposed methods are applied to a pitch-plunge airfoil motion governed by rigid-body dynamics and a one-dimensional viscous Burgers' equation. The current work also demonstrates the prediction capability of various discretized-physics-informed neural networks outside the domain where the data is available or governing equation-based residuals are minimized.

翻译：本研究针对一种物理信息神经网络（PINN）的开发，其损失项源自离散化的时变降阶系统。首先，采用有限差分格式（亦可采用其他离散化技术）对控制方程进行离散化处理，随后通过本征正交分解（POD）-Galerkin方法将其投影至降阶或潜在空间。接着，将离散化降阶方程产生的残差作为额外损失惩罚项，与基于人工神经网络（ANN）、长短期记忆神经网络（LSTM）等不同深度学习变体的数据驱动损失项相结合。LSTM神经网络已在纯数据驱动环境下被证明对时变问题极为有效。本工作进一步展示了LSTM网络在物理信息神经网络（PINN）中相较于ANN网络的潜在优势。采用离散化控制方程而非连续形式的关键优势在于增强了PINN输入端的灵活性。本研究利用从少量、中等到大规模的不同数据规模，评估了在数据极度稀疏或完全缺失情况下离散化物理信息神经网络的潜力。所提方法被应用于由刚体动力学驱动的俯仰-沉浮翼型运动以及一维黏性Burgers方程。本工作同时展示了各类离散化物理信息神经网络在数据可用域或基于控制方程的残差最小化域之外的预测能力。