FunDiff: diffusion models over function spaces for physics-informed generative modeling.

Area of Science:

• Computational Physics
• Generative AI
• Scientific Machine Learning

Background:

• Generative models like diffusion models and flow matching excel at discrete data synthesis (images, videos).
• Adapting these models for continuous functions in physical applications is challenging due to complex governing laws.
• Existing methods struggle to incorporate physical priors and handle varying data discretizations.

Purpose of the Study:

• To introduce FunDiff, an efficient and robust framework for generative modeling in function spaces.
• To enable the synthesis of continuous functions that adhere to physical laws for scientific applications.
• To provide theoretical guarantees and demonstrate practical effectiveness in fluid and solid mechanics.

Main Methods:

• FunDiff combines a latent diffusion process with a function autoencoder architecture.
• Handles input functions with varying discretizations and generates continuous functions.
• Incorporates physical priors via architectural constraints or physics-informed loss functions.

Main Results:

• Theoretically established minimax optimality guarantees for density estimation in function spaces.
• Demonstrated FunDiff's effectiveness in fluid dynamics and solid mechanics applications.
• Generated physically consistent samples with high fidelity, robust to noisy and low-resolution data.

Conclusions:

• FunDiff offers a powerful new approach for generative modeling in scientific domains involving continuous functions.
• The framework successfully integrates physical laws, ensuring generated data is scientifically plausible.
• This work advances the application of generative AI to complex physical systems.