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

Area of Science:

• Computational Physics
• Machine Learning
• Scientific Computing

Background:

• Conventional diffusion models struggle with irregular domains and geometric variability due to grid-based, snapshot-level representations.
• Existing methods lack adaptability for complex physical systems with diverse geometries.

Purpose of the Study:

• To develop a novel point-wise conditional diffusion framework for efficient and generalizable prediction of physical systems with irregular geometries.
• To enable direct operation of the diffusion process on query points over arbitrary geometries, bypassing grid topology and temporal discretization.

Main Methods:

• A point-wise conditional diffusion framework operating directly on query points defined over arbitrary geometries.
• Incorporation of positional encoding to address spectral bias and capture high-frequency physical details and localized geometric variations.
• Utilized denoising diffusion implicit model (DDIM) sampling for efficient inference.

Main Results:

• The point-wise approach significantly outperforms conventional image-based diffusion methods, reducing mean absolute error by 35.8% with 94.4% less training time and 89.0% fewer parameters.
• Achieved error reductions of 53% to 94% compared to established surrogate models (DeepONet, Meshgraphnet) across three distinct physical systems.
• Demonstrated computational scalability in large-scale aerodynamic systems, achieving superior performance with fewer training points and reduced training cost.

Conclusions:

• The proposed point-wise conditional diffusion framework offers a flexible, scalable, and highly accurate solution for physical system prediction across diverse geometries.
• The method achieves near real-time inference with deterministic reproducibility, outperforming existing approaches in both accuracy and efficiency.
• The framework generalizes robustly to unseen geometric configurations and shows significant advantages in training time and parameter efficiency.