Neural PDE operator learning on domains with interesting geometry

Can ML solve PDEs on non-Euclidean domains?

2019-10-15 — 2025-03-07

Wherein implicit diffeomorphisms are learned to map Euclidean grids onto irregular manifolds, and neural operators are adapted to mask and represent partial differential equations on such domains.

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

Placeholder for a discussion of how to apply machine learning to partial differential equations on interesting weirdly-shaped domains, by which I mean things that are not toruses, spheres, or hyper-rectangles.

Why does this even need saying?, you might ask. Surely most PDEs are solved on interestingly-shaped objects? That would indicate that you come from a PDE background. On the ML side, the ascendancy of neural nets means that per default everything is on non-challenging geometries. The field has seen early success on rasterised grids (and spheres and toruses) which are easy to analysis and convenient to compute efficiently. The next step is to generalise to more interesting geometries and consider the implications of these generalisations. That is what this notebook is about.

1 Mapping Euclidean methods to non-Euclidean domains

This seems to be mostly the implicit representation to learn diffeomorphisms to map between a euclidean grid and a non-euclidean domain.

2 Masking

TODO

3 Transformers

TODO

4 Graph neural operators

TODO

5 DeepONets

TODO

6 References

Alkin, Fürst, Schmid, et al. 2024. “Universal Physics Transformers: A Framework For Efficiently Scaling Neural Operators.” In Advances in Neural Information Processing Systems.

Azizzadenesheli, Kovachki, Li, et al. 2024. “Neural Operators for Accelerating Scientific Simulations and Design.” Nature Reviews Physics.

Blechschmidt, and Ernst. 2021. “Three Ways to Solve Partial Differential Equations with Neural Networks — A Review.” GAMM-Mitteilungen.

Boullé, and Townsend. 2023. “A Mathematical Guide to Operator Learning.”

Brandstetter, Worrall, and Welling. 2022. “Message Passing Neural PDE Solvers.” In International Conference on Learning Representations.

Hao, Wang, Su, et al. 2023. “GNOT: A General Neural Operator Transformer for Operator Learning.” In Proceedings of the 40th International Conference on Machine Learning.

Kag, and Gopinath. 2024. “Physics-Informed Neural Network for Modeling Dynamic Linear Elasticity.”

Kossaifi, Kovachki, Li, et al. 2024. “A Library for Learning Neural Operators.”

Kovachki, Li, Liu, et al. 2023. “Neural Operator: Learning Maps Between Function Spaces With Applications to PDEs.” Journal of Machine Learning Research.

Lam, Sanchez-Gonzalez, Willson, et al. 2023. “GraphCast: Learning Skillful Medium-Range Global Weather Forecasting.”

Li, Huang, Liu, et al. 2023. “Fourier Neural Operator with Learned Deformations for PDEs on General Geometries.” Journal of Machine Learning Research.

Li, Kovachki, Azizzadenesheli, Liu, Stuart, et al. 2020. “Multipole Graph Neural Operator for Parametric Partial Differential Equations.” In Advances in Neural Information Processing Systems.

Li, Kovachki, Azizzadenesheli, Liu, Bhattacharya, et al. 2020. “Fourier Neural Operator for Parametric Partial Differential Equations.”

Li, Kovachki, Choy, et al. 2023. “Geometry-Informed Neural Operator for Large-Scale 3D PDEs.” In Proceedings of the 37th International Conference on Neural Information Processing Systems. NIPS ’23.

Liu, Jafarzadeh, and Yu. 2023. “Domain Agnostic Fourier Neural Operators.”

Lu, Meng, Mao, et al. 2021. “DeepXDE: A Deep Learning Library for Solving Differential Equations.” SIAM Review.

Ruhe, Gupta, de Keninck, et al. 2023. “Geometric Clifford Algebra Networks.” In arXiv Preprint arXiv:2302.06594.

White, Berner, Kossaifi, et al. 2023. “Physics-Informed Neural Operators with Exact Differentiation on Arbitrary Geometries.” In.

Wu, Luo, Wang, et al. 2024. “Transolver: A Fast Transformer Solver for PDEs on General Geometries.”

Zhong, and Meidani. 2024. “Physics-Informed Geometry-Aware Neural Operator.”