Machine learning for partial differential equations using diffusion models

Neural physics for neural physics

February 25, 2025 — March 13, 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

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Diffusion models for PDE learning.

Figure 1

The terminology is confusing since we use diffusion models to learn PDEs, but the PDEs often model diffusion processes themselves. Sometimes, the diffusion models doing the modelling aren’t actually diffusive; they’re based on Poisson flow generative models.

Naming things is hell.

PDEs typically introduce manifolds of solution so we might imagine Non-Euclidean diffusion models are useful for PDE learning.

1 Classical diffusion models

TBD

2 Incoming

3 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.
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