Neural learning for spatiotemporal systems
2020-09-15 — 2024-11-13
Wherein neural learning for image-to-image regression of 2D fields is treated; stochastic PDEs are learned via U-Net and operator-learning methods, inverse problems are approached with low-rank or lattice Gaussian processes, and geospatial scales with out-of-sample spatial inference are examined.
dynamical systems
machine learning
neural nets
sciml
SDEs
signal processing
spatial
statistics
stochastic processes
time series
On regression from 2D images to 2D images (sometimes 3D is OK) using neural networks. Useful in learning stochastic partial differential equations and other processes, which use the tools of dynamical NNs and their ilk. Probably handy for machine learning physics and especially PDEs.
In the inverse setting, useful tools might include low rank or lattice Gaussian processes.
I know little about this topic yet. But here are some articles to read.
1 U-Net
A convnet architecture designed for image segmentation but useful in general image->image regression (Ronneberger, Fischer, and Brox 2015; Shelhamer, Long, and Darrell 2017).
2 Operator learning
See ML for PDEs.
3 Geospatial scale
4 Foundation models for
5 References
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Beck, E, and Jentzen. 2019. “Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-Order Backward Stochastic Differential Equations.” Journal of Nonlinear Science.
Dupont, Kim, Eslami, et al. 2022. “From Data to Functa: Your Data Point Is a Function and You Can Treat It Like One.” In Proceedings of the 39th International Conference on Machine Learning.
E, Han, and Jentzen. 2017. “Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations.” Communications in Mathematics and Statistics.
E, and Yu. 2018. “The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems.” Communications in Mathematics and Statistics.
Guth, Mojahed, and Sapsis. 2023. “Evaluation of Machine Learning Architectures on the Quantification Of Epistemic and Aleatoric Uncertainties In Complex Dynamical Systems.” SSRN Scholarly Paper.
Han, Jentzen, and E. 2018. “Solving High-Dimensional Partial Differential Equations Using Deep Learning.” Proceedings of the National Academy of Sciences.
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Lu, Meng, Mao, et al. 2021. “DeepXDE: A Deep Learning Library for Solving Differential Equations.” SIAM Review.
Mahesh, Collins, Bonev, et al. 2024. “Huge Ensembles Part I: Design of Ensemble Weather Forecasts Using Spherical Fourier Neural Operators.”
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Särkkä, Solin, and Hartikainen. 2013. “Spatiotemporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering.” IEEE Signal Processing Magazine.
Shankar, Portwood, Mohan, et al. 2020. “Learning Non-Linear Spatio-Temporal Dynamics with Convolutional Neural ODEs.” In Third Workshop on Machine Learning and the Physical Sciences (NeurIPS 2020).
Shelhamer, Long, and Darrell. 2017. “Fully Convolutional Networks for Semantic Segmentation.” IEEE Transactions on Pattern Analysis and Machine Intelligence.
Solin, and Särkkä. 2013. “Infinite-Dimensional Bayesian Filtering for Detection of Quasiperiodic Phenomena in Spatiotemporal Data.” Physical Review E.
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Tartakovsky, Marrero, Perdikaris, et al. 2018. “Learning Parameters and Constitutive Relationships with Physics Informed Deep Neural Networks.”
Yang, Zhang, and Karniadakis. 2020. “Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations.” SIAM Journal on Scientific Computing.
Zammit-Mangion, Ng, Vu, et al. 2021. “Deep Compositional Spatial Models.” Journal of the American Statistical Association.
Zammit-Mangion, and Wikle. 2020. “Deep Integro-Difference Equation Models for Spatio-Temporal Forecasting.” Spatial Statistics.
Zang, Bao, Ye, et al. 2020. “Weak Adversarial Networks for High-Dimensional Partial Differential Equations.” Journal of Computational Physics.
Zhang, Guo, and Karniadakis. 2020. “Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks.” SIAM Journal on Scientific Computing.
Zhang, Lu, Guo, et al. 2019. “Quantifying Total Uncertainty in Physics-Informed Neural Networks for Solving Forward and Inverse Stochastic Problems.” Journal of Computational Physics.