Scientific machine learning (SciML) methods such as physics-informed neural networks (PINNs) are used to estimate parameters of interest from governing equations and small quantities of data. However, there has been little work in assessing how well PINNs perform for inverse problems across wide ranges of governing equations across the mathematical sciences. We present a new and challenging benchmark problem for inverse PINNs based on a parametric sweep of the 2D Burgers' equation with rotational flow. We show that a novel strategy that alternates between first- and second-order optimization proves superior to typical first-order strategies for estimating parameters. In addition, we propose a novel data-driven method to characterize PINN effectiveness in the inverse setting. PINNs' physics-informed regularization enables them to leverage small quantities of data more efficiently than the data-driven baseline. However, both PINNs and the baseline can fail to recover parameters for highly inviscid flows, motivating the need for further development of PINN methods.

本研究针对物理信息神经网络（PINNs）在逆问题上的表现进行了评估，填补了该领域的研究空白。通过对二维巴格斯方程进行参数扫掠，提出了一种新颖的优化策略，结合一、二阶优化，提高了参数估计的精度。研究发现，PINNs在使用少量数据时，能够更有效地发挥物理信息正则化的优势，但在高度理想流动情况下仍然面临挑战，促进了PINN方法的进一步发展。 

通过物理信息神经网络进行流体流动方程识别