Physics-informed super-resolution and forecasting method based on inaccurate partial differential equations and partial observation

Physics-informed machine learning has emerged as a promising approach for modeling physical systems. However, real-world applications often face significant challenges due to the limitations of partial observations and inaccuracies in governing partial differential equations (PDEs). In this work, we propose a novel physics-informed machine learning method named Physics-Informed method based on Inaccurate PDEs and Partial Observation (PIPO), to overcome the above two challenges under the real-world scenario, which aims to address the problems of super-resolution and forecasting simultaneously in physical systems characterized by partial observations and inaccurate PDEs. The proposed method is motivated by two key considerations. First, despite the inherent inaccuracy of PDEs, the differential terms (such as diffusion and advection terms) contain valuable information that can effectively reduce the hypothesis space, thereby enhancing the model's generalization capability. Second, while the data only provides partial observations, it offers crucial supervised constraints at the observed points. These constraints not only facilitate model optimization but also help prevent the degeneracy of PDE loss, where the PDE loss could yield multiple solutions. PIPO integrates an interpolator, encoder, forecaster, decoder, and parameters learner, which are jointly optimized using data loss and PDE losses to reconstruct high-resolution states and forecast future states using only partial observation data and inaccurate PDEs. We leverage the proposed PIPO method to address a real-world problem in air pollutant concentration fields and wind fields, specifically PM2.5 transport dynamics, which are governed by the advection-diffusion equation with unknown diffusion coefficients and the source term. The results in super-resolution reconstruction, forecasting, and multi-hour forecasting highlight the effectiveness of PIPO in capturing complex spatial and temporal dynamics despite the limitations of partial observations and inaccurate PDEs.