Nonparametrically learning spatiotemporal systems


On learning stochastic partial differential equations and other processes using neural networks, gaussian processes and other differentiable techniques. Uses the tools of dynamical NNs and their ilk. Probably handy for machien learning physics.

I know little about this yet. But here are some links

Tensorflow pdes etc

DeepXDE, TenFEM

https://sciml.ai/ (plus maybe Julaifem)?

Arridge, Simon, Peter Maass, Ozan Öktem, and Carola-Bibiane Schönlieb. 2019. “Solving Inverse Problems Using Data-Driven Models.” Acta Numerica 28 (May): 1–174. https://doi.org/10.1017/S0962492919000059.

Beck, Christian, Weinan E, and Arnulf 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 29 (4): 1563–1619. https://doi.org/10.1007/s00332-018-9525-3.

Han, Jiequn, Arnulf Jentzen, and Weinan E. 2018. “Solving High-Dimensional Partial Differential Equations Using Deep Learning.” Proceedings of the National Academy of Sciences 115 (34): 8505–10. https://doi.org/10.1073/pnas.1718942115.

He, QiZhi, David Barajas-Solano, Guzel Tartakovsky, and Alexandre M. Tartakovsky. 2020. “Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport.” Advances in Water Resources 141 (July): 103610. https://doi.org/10.1016/j.advwatres.2020.103610.

Laloy, Eric, and Diederik Jacques. 2019. “Emulation of CPU-Demanding Reactive Transport Models: A Comparison of Gaussian Processes, Polynomial Chaos Expansion, and Deep Neural Networks.” Computational Geosciences 23 (5): 1193–1215. https://doi.org/10.1007/s10596-019-09875-y.

Lu, Lu, Zhiping Mao, and Xuhui Meng. 2019. “DeepXDE: A Deep Learning Library for Solving Differential Equations.” In, 6. http://arxiv.org/abs/1907.04502.

Mo, Shaoxing, Nicholas Zabaras, Xiaoqing Shi, and Jichun Wu. 2019. “Deep Autoregressive Neural Networks for High-Dimensional Inverse Problems in Groundwater Contaminant Source Identification.” Water Resources Research 55 (5): 3856–81. https://doi.org/10.1029/2018WR024638.

Nabian, Mohammad Amin, and Hadi Meidani. 2019. “A Deep Learning Solution Approach for High-Dimensional Random Differential Equations.” Probabilistic Engineering Mechanics 57 (July): 14–25. https://doi.org/10.1016/j.probengmech.2019.05.001.

Patraucean, Viorica, Ankur Handa, and Roberto Cipolla. 2015. “Spatio-Temporal Video Autoencoder with Differentiable Memory,” November. http://arxiv.org/abs/1511.06309.

Raissi, Maziar, Alireza Yazdani, and George Em Karniadakis. 2020. “Hidden Fluid Mechanics: Learning Velocity and Pressure Fields from Flow Visualizations.” Science 367 (6481): 1026–30. https://doi.org/10.1126/science.aaw4741.

Raissi, M., P. Perdikaris, and G. E. Karniadakis. 2019. “Physics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations.” Journal of Computational Physics 378 (February): 686–707. https://doi.org/10.1016/j.jcp.2018.10.045.

Ruthotto, Lars, and Eldad Haber. 2018. “Deep Neural Networks Motivated by Partial Differential Equations,” April. http://arxiv.org/abs/1804.04272.

Särkkä, Simo, and Jouni Hartikainen. 2012. “Infinite-Dimensional Kalman Filtering Approach to Spatio-Temporal Gaussian Process Regression.” In Artificial Intelligence and Statistics. http://www.jmlr.org/proceedings/papers/v22/sarkka12.html.

Särkkä, Simo, A. Solin, and J. Hartikainen. 2013. “Spatiotemporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering.” IEEE Signal Processing Magazine 30 (4, 4): 51–61. https://doi.org/10.1109/MSP.2013.2246292.

Solin, Arno, and Simo Särkkä. 2013. “Infinite-Dimensional Bayesian Filtering for Detection of Quasiperiodic Phenomena in Spatiotemporal Data.” Physical Review E 88 (5): 052909. https://doi.org/10.1103/PhysRevE.88.052909.

Tait, Daniel J., and Theodoros Damoulas. 2020. “Variational Autoencoding of PDE Inverse Problems,” June. http://arxiv.org/abs/2006.15641.

Tartakovsky, Alexandre M., Carlos Ortiz Marrero, Paris Perdikaris, Guzel D. Tartakovsky, and David Barajas-Solano. 2018. “Learning Parameters and Constitutive Relationships with Physics Informed Deep Neural Networks,” August. https://arxiv.org/abs/1808.03398v2.

Yang, Liu, Dongkun Zhang, and George Em Karniadakis. 2020. “Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations.” SIAM Journal on Scientific Computing 42 (1): A292–A317. https://doi.org/10.1137/18M1225409.

Zammit-Mangion, Andrew, and Christopher K. Wikle. 2020. “Deep Integro-Difference Equation Models for Spatio-Temporal Forecasting.” Spatial Statistics 37 (June): 100408. https://doi.org/10.1016/j.spasta.2020.100408.

Zang, Yaohua, Gang Bao, Xiaojing Ye, and Haomin Zhou. 2020. “Weak Adversarial Networks for High-Dimensional Partial Differential Equations.” Journal of Computational Physics 411 (June): 109409. https://doi.org/10.1016/j.jcp.2020.109409.

Zhang, Dongkun, Ling Guo, and George Em Karniadakis. 2020. “Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks.” SIAM Journal on Scientific Computing 42 (2): A639–A665. https://doi.org/10.1137/19M1260141.

Zhang, Dongkun, Lu Lu, Ling Guo, and George Em Karniadakis. 2019. “Quantifying Total Uncertainty in Physics-Informed Neural Networks for Solving Forward and Inverse Stochastic Problems.” Journal of Computational Physics 397 (November): 108850. https://doi.org/10.1016/j.jcp.2019.07.048.