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 uses the tools of dynamical NNs and their ilk. Probably handy for machine learning physics and especially PDEs.

In the inverse setting useful tolls might in addition be low rank or lattice Gaussian processes.

I know little about this topic yet. But here are some articles to read.

## 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).

## Fourier operator learning

See ML for PDES.

## References

Arridge, Simon, Peter Maass, Ozan Γktem, and Carola-Bibiane SchΓΆnlieb. 2019. βSolving Inverse Problems Using Data-Driven Models.β

*Acta Numerica*28 (May): 1β174.Ayed, Ibrahim, and Emmanuel de BΓ©zenac. 2019. βLearning Dynamical Systems from Partial Observations.β In

*Advances In Neural Information Processing Systems*, 12.Beck, Christian, Weinan E, and Arnulf Jentzen. 2019. βMachine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-Order Backward Stochastic Differential Equations.β

*Journal of Nonlinear Science*29 (4): 1563β1619.Dupont, Emilien, Hyunjik Kim, S. M. Ali Eslami, Danilo Rezende, and Dan Rosenbaum. 2022. βFrom Data to Functa: Your Data Point Is a Function and You Can Treat It Like One.β arXiv.

E, Weinan, Jiequn Han, and Arnulf Jentzen. 2017. βDeep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations.β

*Communications in Mathematics and Statistics*5 (4): 349β80.E, Weinan, and Bing Yu. 2018. βThe Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems.β

*Communications in Mathematics and Statistics*6 (1): 1β12.Han, Jiequn, Arnulf Jentzen, and Weinan E. 2018. βSolving High-Dimensional Partial Differential Equations Using Deep Learning.β

*Proceedings of the National Academy of Sciences*115 (34): 8505β10.He, QiZhi, David Barajas-Solano, Guzel Tartakovsky, and Alexandre M. Tartakovsky. 2020. βPhysics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport.β

*Advances in Water Resources*141 (July): 103610.Laloy, Eric, and Diederik Jacques. 2019. βEmulation of CPU-Demanding Reactive Transport Models: A Comparison of Gaussian Processes, Polynomial Chaos Expansion, and Deep Neural Networks.β

*Computational Geosciences*23 (5): 1193β1215.Lu, Lu, Xuhui Meng, Zhiping Mao, and George Em Karniadakis. 2021. βDeepXDE: A Deep Learning Library for Solving Differential Equations.β

*SIAM Review*63 (1): 208β28.Mo, Shaoxing, Nicholas Zabaras, Xiaoqing Shi, and Jichun Wu. 2019. βDeep Autoregressive Neural Networks for High-Dimensional Inverse Problems in Groundwater Contaminant Source Identification.β

*Water Resources Research*55 (5): 3856β81.MΓΌller, Johannes, and Marius Zeinhofer. 2020. βDeep Ritz Revisited.β arXiv.

Nabian, Mohammad Amin, and Hadi Meidani. 2019. βA Deep Learning Solution Approach for High-Dimensional Random Differential Equations.β

*Probabilistic Engineering Mechanics*57 (July): 14β25.Park, Ji Hwan, Shinjae Yoo, and Balu Nadiga. 2019. βMachine Learning Climate Variability.β In, 5.

Patraucean, Viorica, Ankur Handa, and Roberto Cipolla. 2015. βSpatio-Temporal Video Autoencoder with Differentiable Memory.β

*arXiv:1511.06309 [Cs]*, November.Rackauckas, Chris, Alan Edelman, Keno Fischer, Mike Innes, Elliot Saba, Viral B Shah, and Will Tebbutt. 2020. βGeneralized Physics-Informed Learning Through Language-Wide Differentiable Programming.β

*MIT Web Domain*, 6.Raissi, Maziar, P. Perdikaris, and George Em Karniadakis. 2019. βPhysics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations.β

*Journal of Computational Physics*378 (February): 686β707.Raissi, Maziar, Alireza Yazdani, and George Em Karniadakis. 2020. βHidden Fluid Mechanics: Learning Velocity and Pressure Fields from Flow Visualizations.β

*Science*367 (6481): 1026β30.Ronneberger, Olaf, Philipp Fischer, and Thomas Brox. 2015. βU-Net: Convolutional Networks for Biomedical Image Segmentation.β Edited by Nassir Navab, Joachim Hornegger, William M. Wells, and Alejandro F. Frangi.

*Medical Image Computing and Computer-Assisted Intervention β MICCAI 2015*. Lecture Notes in Computer Science. Cham: Springer International Publishing.Ruthotto, Lars, and Eldad Haber. 2018. βDeep Neural Networks Motivated by Partial Differential Equations.β

*arXiv:1804.04272 [Cs, Math, Stat]*, April.SΓ€rkkΓ€, Simo. 2011. βLinear Operators and Stochastic Partial Differential Equations in Gaussian Process Regression.β In

*Artificial Neural Networks and Machine Learning β ICANN 2011*, edited by Timo Honkela, WΕodzisΕaw Duch, Mark Girolami, and Samuel Kaski, 6792:151β58. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer.SΓ€rkkΓ€, Simo, and Jouni Hartikainen. 2012. βInfinite-Dimensional Kalman Filtering Approach to Spatio-Temporal Gaussian Process Regression.β In

*Artificial Intelligence and Statistics*.SΓ€rkkΓ€, Simo, A. Solin, and J. Hartikainen. 2013. βSpatiotemporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering.β

*IEEE Signal Processing Magazine*30 (4): 51β61.Shankar, Varun, Gavin D Portwood, Arvind T Mohan, Peetak P Mitra, Christopher Rackauckas, Lucas A Wilson, David P Schmidt, and Venkatasubramanian Viswanathan. 2020. βLearning Non-Linear Spatio-Temporal Dynamics with Convolutional Neural ODEs.β In

*Third Workshop on Machine Learning and the Physical Sciences (NeurIPS 2020)*.Shelhamer, Evan, Jonathan Long, and Trevor Darrell. 2017. βFully Convolutional Networks for Semantic Segmentation.β

*IEEE Transactions on Pattern Analysis and Machine Intelligence*39 (4): 640β51.Solin, Arno, and Simo SΓ€rkkΓ€. 2013. βInfinite-Dimensional Bayesian Filtering for Detection of Quasiperiodic Phenomena in Spatiotemporal Data.β

*Physical Review E*88 (5): 052909.Tait, Daniel J., and Theodoros Damoulas. 2020. βVariational Autoencoding of PDE Inverse Problems.β

*arXiv:2006.15641 [Cs, Stat]*, June.Tartakovsky, Alexandre M., Carlos Ortiz Marrero, Paris Perdikaris, Guzel D. Tartakovsky, and David Barajas-Solano. 2018. βLearning Parameters and Constitutive Relationships with Physics Informed Deep Neural Networks,β August.

Yang, Liu, Dongkun Zhang, and George Em Karniadakis. 2020. βPhysics-Informed Generative Adversarial Networks for Stochastic Differential Equations.β

*SIAM Journal on Scientific Computing*42 (1): A292β317.Zammit-Mangion, Andrew, Tin Lok James Ng, Quan Vu, and Maurizio Filippone. 2021. βDeep Compositional Spatial Models.β

*Journal of the American Statistical Association*0 (0): 1β22.Zammit-Mangion, Andrew, and Christopher K. Wikle. 2020. βDeep Integro-Difference Equation Models for Spatio-Temporal Forecasting.β

*Spatial Statistics*37 (June): 100408.Zang, Yaohua, Gang Bao, Xiaojing Ye, and Haomin Zhou. 2020. βWeak Adversarial Networks for High-Dimensional Partial Differential Equations.β

*Journal of Computational Physics*411 (June): 109409.Zhang, Dongkun, Ling Guo, and George Em Karniadakis. 2020. βLearning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks.β

*SIAM Journal on Scientific Computing*42 (2): A639β65.Zhang, Dongkun, Lu Lu, Ling Guo, and George Em Karniadakis. 2019. βQuantifying Total Uncertainty in Physics-Informed Neural Networks for Solving Forward and Inverse Stochastic Problems.β

*Journal of Computational Physics*397 (November): 108850.
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