Machine learning for partial differential equations using diffusion models

Neural physics for neural physics

February 25, 2025 — February 25, 2025

calculus
dynamical systems
geometry
Hilbert space
how do science
Lévy processes
machine learning
neural nets
PDEs
physics
regression
sciml
SDEs
signal processing
statistics
statmech
stochastic processes
surrogate
time series
uncertainty

Suspiciously similar content

Diffusion models for PDE learning.

Figure 1

Slightly confusing terminology, because we are using diffusion models to learn PDEs, but the PDEs themselves are often used to model diffusion processes. Also sometimes the diffusion models that do the modelling aren’t actually diffusive, but are based on Poisson flow generative models.

Naming things is hell.

1 Classical diffusion models

TBD

2 Poisson Flow generative models

These are based on non-diffusive physics but also seem to be used to simulate physics (Xu et al. 2022, 2023).

Popsci explaantion in Quanta magazine.

3 Incoming

4 References

Bastek, Sun, and Kochmann. 2024. “Physics-Informed Diffusion Models.”
Kita, Dubiński, Rokita, et al. 2024. “Generative Diffusion Models for Fast Simulations of Particle Collisions at CERN.”
Lim, Kovachki, Baptista, et al. 2023. “Score-Based Diffusion Models in Function Space.”
Liu, Luo, Xu, et al. 2023. “GenPhys: From Physical Processes to Generative Models.”
Seo, Um, Ye, et al. 2024. “Physics-Guided Diffusion Models for Inverse Design.” In 2024 Conference on Lasers and Electro-Optics Pacific Rim (CLEO-PR).
Shi, Yan, Guo, et al. 2024. “Multi-Physics Simulation Guided Generative Diffusion Models with Applications in Fluid and Heat Dynamics.”
Shu, Li, and Barati Farimani. 2023. “A Physics-Informed Diffusion Model for High-Fidelity Flow Field Reconstruction.” Journal of Computational Physics.
Xu, Liu, Tegmark, et al. 2022. “Poisson Flow Generative Models.” In Proceedings of the 36th International Conference on Neural Information Processing Systems. NIPS ’22.
Xu, Liu, Tian, et al. 2023. “PFGM++: Unlocking the Potential of Physics-Inspired Generative Models.” In Proceedings of the 40th International Conference on Machine Learning. ICML’23.
Yuan, Song, Iqbal, et al. 2023. “PhysDiff: Physics-Guided Human Motion Diffusion Model.”
Zhang, Yan, Perelli, et al. 2024. “Phy-Diff: Physics-Guided Hourglass Diffusion Model for Diffusion MRI Synthesis.” In Medical Image Computing and Computer Assisted Intervention – MICCAI 2024.