Point-wise conditional diffusion models for physical systems with shape variations: Applications to spatio-temporal...

Conventional diffusion models for physical system prediction rely on grid-based and snapshot-level representations, limiting their adaptability to irregular domains and geometric variability. This study introduces a novel point-wise conditional diffusion framework that enables efficient and generalizable prediction of complex physical systems with diverse and irregular geometries. The core idea enables the diffusion process to operate directly on query points defined over arbitrary geometries, in contrast to conventional diffusion models that apply denoising to an entire snapshot at once. Each query point is independently conditioned on its spatio-temporal coordinates and physical information, allowing point-wise modeling without relying on grid topology or temporal discretization. To address the spectral bias inherent in coordinate-based representations, positional encoding is incorporated to capture high-frequency physical details and localized geometric variations. The flexibility and scalability of the proposed framework enable it to generalize across three physical domains: two-dimensional spatio-temporal systems and a three-dimensional large-scale aerodynamic system, without requiring additional preprocessing. Experimental results demonstrate that the proposed method, employing denoising diffusion implicit model (DDIM) sampling with only 5-10 steps, enables near real-time inference while ensuring deterministic reproducibility essential for physical system prediction. Comparative analysis reveals that our point-wise approach outperforms conventional image-based diffusion methods, yielding 35.8% reduction in mean absolute error with 94.4% less training time and 89.0% fewer parameters. Performance evaluations across three distinct physical systems consistently demonstrate superior accuracy, with error reductions ranging from 53% to 94% compared to established data-flexible surrogate models including DeepONet and Meshgraphnet. Furthermore, the framework demonstrates remarkable computational scalability in large-scale automotive aerodynamic systems, achieving superior performance with only 50% of training points while significantly reducing training cost, and generalizes robustly to previously unseen geometric configurations.