We propose a physics-constrained convolutional neural network (PC-CNN) to solve two types of inverse problems in partial differential equations (PDEs), which are nonlinear and vary both in space and time. In the first inverse problem, we are given data that is offset by spatially varying systematic error (i.e., the bias, also known as the epistemic uncertainty). The task is to uncover the true state, which is the solution of the PDE, from the biased data. In the second inverse problem, we are given sparse information on the solution of a PDE. The task is to reconstruct the solution in space with high-resolution. First, we present the PC-CNN, which constrains the PDE with a time-windowing scheme to handle sequential data. Second, we analyse the performance of the PC-CNN for uncovering solutions from biased data. We analyse both linear and nonlinear convection-diffusion equations, and the Navier-Stokes equations, which govern the spatiotemporally chaotic dynamics of turbulent flows. We find that the PC-CNN correctly recovers the true solution for a variety of biases, which are parameterised as non-convex functions. Third, we analyse the performance of the PC-CNN for reconstructing solutions from sparse information for the turbulent flow. We reconstruct the spatiotemporal chaotic solution on a high-resolution grid from only < 1\% of the information contained in it. For both tasks, we further analyse the Navier-Stokes solutions. We find that the inferred solutions have a physical spectral energy content, whereas traditional methods, such as interpolation, do not. This work opens opportunities for solving inverse problems with partial differential equations.

翻译：我们提出了一种物理约束卷积神经网络（PC-CNN），用于求解偏微分方程（PDE）中两类非线性的时空变化反问题。在第一个反问题中，给定数据受到空间变化的系统误差（即偏差，也称为认知不确定性）影响。任务是从带有偏差的数据中还原出作为PDE解的真实状态。在第二个反问题中，给定PDE解的稀疏信息。任务是在空间上高分辨率地重建该解。首先，我们提出PC-CNN，该网络通过时间窗口化方案约束PDE以处理序列数据。其次，我们分析了PC-CNN从偏差数据中还原解的性能。我们分析了线性和非线性对流-扩散方程，以及描述湍流时空混沌动力学的Navier-Stokes方程。我们发现，对于参数化为非凸函数的多种偏差，PC-CNN均能正确恢复真实解。第三，我们分析了PC-CNN从稀疏信息重建湍流解的性能。我们仅利用解中所含信息的<1%，就在高分辨率网格上重建了时空混沌解。针对这两项任务，我们进一步分析了Navier-Stokes方程的解。我们发现推断出的解具有物理意义上的谱能量分布，而传统方法（如插值）则不具备这一特性。这项工作为解决偏微分方程反问题开辟了新途径。