Machine learning for inverting partial differential equations

2017-05-15 — 2025-04-17

Wherein tomography through partial differential equations is undertaken by learned solution operators and neural priors, and computational inversion is accelerated by data-driven surrogate models.

calculus

dynamical systems

geometry

Hilbert space

how do science

machine learning

neural nets

PDEs

physics

regression

sciml

SDEs

signal processing

statistics

statmech

stochastic processes

surrogate

time series

uncertainty

Tomography through PDEs via machine learning

See Inverse problems in PDEs for the classical setting. What extra do we get from ML? I have nothing to say right now; but check the references.

1 References

Adler, and Öktem. 2017. “Solving Ill-Posed Inverse Problems Using Iterative Deep Neural Networks.” Inverse Problems.

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

Chen, Cheng, Feng, et al. 2020. “Physics-Informed Neural Networks for Inverse Problems in Nano-Optics and Metamaterials.” Optics Express.

Cockayne, Oates, Sullivan, et al. 2016. “Probabilistic Numerical Methods for Partial Differential Equations and Bayesian Inverse Problems.”

———, et al. 2017. “Probabilistic Numerical Methods for PDE-Constrained Bayesian Inverse Problems.” In AIP Conference Proceedings.

Frerix, Kochkov, Smith, et al. 2021. “Variational Data Assimilation with a Learned Inverse Observation Operator.” In.

Ghattas, and Willcox. 2021. “Learning Physics-Based Models from Data: Perspectives from Inverse Problems and Model Reduction.” Acta Numerica.

Haitsiukevich, Poyraz, Marttinen, et al. 2024. “Diffusion Models as Probabilistic Neural Operators for Recovering Unobserved States of Dynamical Systems.”

Jin, McCann, Froustey, et al. 2017. “Deep Convolutional Neural Network for Inverse Problems in Imaging.” IEEE Transactions on Image Processing.

Kadeethum, O’Malley, Fuhg, et al. 2021. “A Framework for Data-Driven Solution and Parameter Estimation of PDEs Using Conditional Generative Adversarial Networks.” NPJ Computational Materials.

Lu, Jin, Pang, et al. 2021. “Learning Nonlinear Operators via DeepONet Based on the Universal Approximation Theorem of Operators.” Nature Machine Intelligence.

MacKinlay, Pagendam, Kuhnert, et al. 2021. “Model Inversion for Spatio-Temporal Processes Using the Fourier Neural Operator.” In.

Psaros, Meng, Zou, et al. 2023. “Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons.” Journal of Computational Physics.

Raissi, Perdikaris, and 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.

Tait, and Damoulas. 2020. “Variational Autoencoding of PDE Inverse Problems.” arXiv:2006.15641 [Cs, Stat].

Yang, and Perdikaris. 2019. “Adversarial Uncertainty Quantification in Physics-Informed Neural Networks.” Journal of Computational Physics.

Zhu, Zabaras, Koutsourelakis, et al. 2019. “Physics-Constrained Deep Learning for High-Dimensional Surrogate Modeling and Uncertainty Quantification Without Labeled Data.” Journal of Computational Physics.