We present a new physics-informed machine learning approach for the inversion of partial differential equation (PDE) models with heterogeneous parameters. In our approach, the space-dependent partially observed parameters and states are approximated via Karhunen-Loève expansions (KLEs). Each of these KLEs is then conditioned on their corresponding measurements, resulting in low-dimensional models of the parameters and states that resolve observed data. Finally, the coefficients of the KLEs are estimated by minimizing the norm of the residual of the PDE model evaluated at a finite set of points in the computational domain, ensuring that the reconstructed parameters and states are consistent with both the observations and the PDE model to an arbitrary level of accuracy.

In our approach, KLEs are constructed using the eigendecomposition of covariance models of spatial variability. For the model parameters, we employ a parameterized covariance model calibrated on parameter observations; for the model states, the covariance is estimated from a number of forward simulations of the PDE model corresponding to realizations of the parameters drawn from their KLE. We apply the proposed approach to identifying heterogeneous log-diffusion coefficients in diffusion equations from spatially sparse measurements of the log-diffusion coefficient and the solution of the diffusion equation. We find that the proposed approach compares favorably against state-of-the-art point estimates such as maximum a posteriori estimation and physics-informed neural networks.

我们提出了一种具有异质性参数的偏微分方程（PDE）模型反演的新型物理信息机器学习方法。在我们的方法中，通过Karhunen-Loève展开（KLE）近似于空间相关的部分观测参数​​和状态。然后，将这些KLE中的每一个均以其相应的测量为条件，从而生成可解析观测数据的参数和状态的低维模型。最后，通过最小化在计算域中有限点集上评估的PDE模型的残差范数来估计KLE的系数，从而确保重构的参数和状态与观测值和PDE模型一致任意级别的准确性。

在我们的方法中，使用空间变异性的协方差模型的特征分解来构造KLE。对于模型参数，我们采用根据参数观测值校准的参数化协方差模型；对于模型状态，协方差是根据PDE模型的多个正向模拟估算的，该模拟对应于从其KLE提取的参数的实现。我们将提出的方法用于从对数扩散系数的空间稀疏测量和扩散方程的解中识别扩散方程中的非均质对数扩散系数。我们发现，提出的方法与最新的点估计（例如最大后验估计和物理信息神经网络）相比具有优势。