Taming transformers to learn high-resolution partial differential equation distributions via conditional latent diffusion priors

Obtaining the solution space for complex physical systems presents a significant challenge. Traditionally, quantifying the flow dynamics has been based on numerical calculation, which requires significant computational resources and time, restricting its use in further applications. Recent advances in machine learning have produced promising surrogate models with less forecast error and faster prediction. These complex, unsteady dynamics are difficult to represent with a single mean solution. For many practical scenarios, obtaining the full distribution of possible states is essential. Here, we introduce a Transformer-based latent diffusion model with conditional injection that enables direct learning of spatiotemporal states from the full distribution, given its physical parameters and initial condition. The effectiveness and scalability of the proposed framework are evaluated through two case studies at different resolutions. We further validate the model's noise robustness and stability via perturbation experiments with noise contamination and low-resolution inputs. The results show that the framework exhibits strong robustness, generating accurate flow sequences under moderate noise and low-resolution conditions while significantly reducing computational costs. Moreover, its inherent scalability provides a valuable reference for developing large-scale fluid dynamics models.