Machine learning for partial differential equations via flows

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

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Normalising flows for PDE learning.

Figure 1

Lipman et al. (2023) seems to be the origin point, extended by Kerrigan, Migliorini, and Smyth (2024) to function-valued PDEs.

Figure 2: An illustration of our FFM method. The vector field \(v_t(f) \in \mathcal{F}\) (in black) transforms a noise sample \(g \sim \mu_0=\mathcal{N}\left(0, C_0\right)\) drawn from a Gaussian process with a Matérn kernel (at \(t=0\) ) to the function \(f(x)=\sin (x)\) (at \(t=1\) ) via solving a function space ODE. By sampling many such \(g \sim \mu_0\), we define a conditional path of measures \(\mu_t^f\) approximately interpolating between \(\mathcal{N}\left(0, C_0\right)\) and the function \(f\), which we marginalize over samples \(f \sim \nu\) from the data distribution in order to obtain a path of measures approximately interpolating between \(\mu_0\) and \(\nu\). (Kerrigan, Migliorini, and Smyth 2024)

1 References

Cheng, Han, Maddix, et al. 2024. “Hard Constraint Guided Flow Matching for Gradient-Free Generation of PDE Solutions.”
Kerrigan, Migliorini, and Smyth. 2024. “Functional Flow Matching.” In Proceedings of The 27th International Conference on Artificial Intelligence and Statistics.
———. 2025. “Dynamic Conditional Optimal Transport Through Simulation-Free Flows.” In Advances in Neural Information Processing Systems.
Lipman, Chen, Ben-Hamu, et al. 2023. “Flow Matching for Generative Modeling.” In.
Liu, Gong, and Liu. 2022. “Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow.” In.
Shi, Gao, Ross, et al. 2024. “Universal Functional Regression with Neural Operator Flows.” In Transactions on Machine Learning Research.