Zijie Li

Zijie Li, Anthony Zhou, Amir Barati Farimani · cmu

Autoregressive next-step prediction models have become the de-facto standard for building data-driven neural solvers to forecast time-dependent partial differential equations (PDEs). Denoise training that is closely related to diffusion probabilistic model has been shown to enhance the temporal stability of neural solvers, while its stochastic inference mechanism enables ensemble predictions and uncertainty quantification. In principle, such training involves sampling a series of discretized diffusion timesteps during both training and inference, inevitably increasing computational overhead. In addition, most diffusion models apply isotropic Gaussian noise on structured, uniform grids, limiting their adaptability to irregular domains. We propose a latent diffusion model for PDE simulation that embeds the PDE state in a lower-dimensional latent space, which significantly reduces computational costs. Our framework uses an autoencoder to map different types of meshes onto a unified structured latent grid, capturing complex geometries. By analyzing common diffusion paths, we propose to use a coarsely sampled noise schedule from flow matching for both training and testing. Numerical experiments show that the proposed model outperforms several deterministic baselines in both accuracy and long-term stability, highlighting the potential of diffusion-based approaches for robust data-driven PDE learning.

Dule Shu, Wilson Zhen, Zijie Li et al.

Fluid data completion is a research problem with high potential benefit for both experimental and computational fluid dynamics. An effective fluid data completion method reduces the required number of sensors in a fluid dynamics experiment, and allows a coarser and more adaptive mesh for a Computational Fluid Dynamics (CFD) simulation. However, the ill-posed nature of the fluid data completion problem makes it prohibitively difficult to obtain a theoretical solution and presents high numerical uncertainty and instability for a data-driven approach (e.g., a neural network model). To address these challenges, we leverage recent advancements in computer vision, employing the vector quantization technique to map both complete and incomplete fluid data spaces onto discrete-valued lower-dimensional representations via a two-stage learning procedure. We demonstrated the effectiveness of our approach on Kolmogorov flow data (Reynolds number: 1000) occluded by masks of different size and arrangement. Experimental results show that our proposed model consistently outperforms benchmark models under different occlusion settings in terms of point-wise reconstruction accuracy as well as turbulent energy spectrum and vorticity distribution.