Data Imputation with Uncertainty Using Stochastic Physics-Informed Learning

The influx of uncertain generations and loads like Electric Vehicle (EV)s challenges the grid stability, demanding dynamic data for analysis, e.g., data from a few Measurement Units (PMUs). To handle data scarcity with uncertainty, we propose to interpolate the Low-Resolution (LR) meters. Existing methods are mainly deterministic and based on correlations. Hence, we propose to model the process as Stochastic Differential Equations (SDEs) and gain an advanced interpolator by combining advanced deep learning techniques with SDE constraints. Specifically, we propose a Stochastic Physics-Informed Neural Network (SPINN) that can incorporate both PMU and LR data for interpolation. SPINN is composed of an expected function-form loss and a distribution loss for the boundary condition. To measure the distributional loss, traditional methods utilize GAN, which learns a distributional measure with a discriminator. However, we observe that using GAN requires extremely high capacity for the discriminator to evaluate the distribution of trajectories. To address the issue, we replace the discriminator with a distribution measure, Maximum Mean Discrepancy (MMD), effective for high-dimensional samples with a simple joint distribution. Finally, we show our MMD-SPINN can generate more accurate trajectories for LR meters than cutting-edge GAN models.