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.

翻译：自然现象与工程系统的信息通常蕴含于数据中，但数据可能因模型与实验中的系统误差而受到污染。本文提出了一种工具，通过从数据中移除系统误差，揭示潜在物理系统的时空解。该工具为物理约束卷积神经网络（PC-CNN），它融合了系统控制方程与数据的双重信息。我们聚焦于由偏微分方程建模的基础现象，如线性对流、Burgers方程及二维湍流。首先，我们定义问题，描述物理约束卷积神经网络并参数化系统误差；其次，从受多模态系统误差污染的数据中挖掘解决方案；第三，针对不同系统误差开展参数化研究，证明方法的稳健性；第四，分析所挖掘解的物理特性，表明PC-CNN推断的解具有物理合理性，而受系统误差污染的数据无法满足控制方程。本研究为消除模型中的认知误差与测量中的系统误差开辟了新路径。