Qi Wang

Mengtao Yan, Qi Wang, Haining Wang et al.

Simulation of fluid flows is crucial for modeling physical phenomena like meteorology, aerodynamics, and biomedicine. Classical numerical solvers often require fine spatiotemporal grids to satisfy stability, consistency, and convergence conditions, leading to substantial computational costs. Although machine learning has demonstrated better efficiency, they typically suffer from issues of interpretability, generalizability, and data dependency. Hence, we propose a learnable and differentiable finite volume solver, called LDSolver, designed for efficient and accurate simulation of fluid flows on spatiotemporal coarse grids. LDSolver comprises two key components: (1) a differentiable finite volume solver, and (2) an learnable module providing equivalent approximation for fluxes (derivatives and interpolations), and temporal error correction on coarse grids. Even with limited training data (e.g., only a few trajectories), our model could accelerate the simulation while maintaining a high accuracy with superior generalizability. Experiments on different flow systems (e.g., Burgers, decaying, forced and shear flows) show that LDSolver achieves state-of-the-art performance, surpassing baseline models with notable margins.

Han Wan, Rui Zhang, Qi Wang et al.

Accurately modeling and forecasting complex systems governed by partial differential equations (PDEs) is crucial in various scientific and engineering domains. However, traditional numerical methods struggle in real-world scenarios due to incomplete or unknown physical laws. Meanwhile, machine learning approaches often fail to generalize effectively when faced with scarce observational data and the challenge of capturing local and global features. To this end, we propose the Physics-encoded Spectral Attention Network (PeSANet), which integrates local and global information to forecast complex systems with limited data and incomplete physical priors. The model consists of two key components: a physics-encoded block that uses hard constraints to approximate local differential operators from limited data, and a spectral-enhanced block that captures long-range global dependencies in the frequency domain. Specifically, we introduce a novel spectral attention mechanism to model inter-spectrum relationships and learn long-range spatial features. Experimental results demonstrate that PeSANet outperforms existing methods across all metrics, particularly in long-term forecasting accuracy, providing a promising solution for simulating complex systems with limited data and incomplete physics.