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 from the biased data the true state, which is the solution of the PDE. 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 simple 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 biased data for the turbulent flow. We reconstruct the spatiotemporal chaotic solution on a high-resolution grid from only 2\% 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：我们提出一种物理约束卷积神经网络（PC-CNN），用于求解两类非线性且随空间时间变化的偏微分方程（PDE）反问题。第一类反问题中，输入数据包含由空间变化的系统性误差（即偏差，也称为认知不确定性）产生的偏移。任务是从带偏数据中恢复真实状态——即偏微分方程的解。第二类反问题中，仅给定偏微分方程解的稀疏信息，任务是在空间上高分辨率重构该解。首先，我们提出PC-CNN，通过简单的时间窗口方案约束偏微分方程以处理序列数据。其次，分析PC-CNN从带偏数据中恢复解的性能，针对线性与非线性对流扩散方程，以及控制湍流时空混沌动力学的纳维-斯托克斯方程。研究发现，对于多种参数化为非凸函数的偏差，PC-CNN能正确恢复真实解。第三，分析PC-CNN从湍流带偏数据重构解的性能。仅利用解中2%的信息，便能在高分辨率网格上重构时空混沌解。针对两类任务，我们进一步分析纳维-斯托克斯方程的解，发现推断解具有物理谱能量分布，而传统方法（如插值）则不具备。该工作为求解偏微分方程反问题开辟了新途径。