Information on natural phenomena and engineering systems is typically contained in data. Data can be corrupted by systematic errors in models and experiments. In this paper, we propose a tool to uncover the spatiotemporal solution of the underlying physical system by removing the systematic errors from data. The tool is the physics-constrained convolutional neural network (PC-CNN), which combines information from both the systems governing equations and data. We focus on fundamental phenomena that are modelled by partial differential equations, such as linear convection, Burgers equation, and two-dimensional turbulence. First, we formulate the problem, describe the physics-constrained convolutional neural network, and parameterise the systematic error. Second, we uncover the solutions from data corrupted by large multimodal systematic errors. Third, we perform a parametric study for different systematic errors. We show that the method is robust. Fourth, we analyse the physical properties of the uncovered solutions. We show that the solutions inferred from the PC-CNN are physical, in contrast to the data corrupted by systematic errors that does not fulfil the governing equations. This work opens opportunities for removing epistemic errors from models, and systematic errors from measurements.

该研究提出了一种名为PC-CNN的物理约束卷积神经网络来发现基于偏微分方程的物理系统的空间和时间解，并从受到大量多模式系统误差损坏的数据中揭示解决方案，该方法是强大且灵活的。

基于物理约束的卷积神经网络应用于数据系统误差的求解