# Learning continuous-time PDEs from sparse data with graph neural networks

@article{Iakovlev2021LearningCP, title={Learning continuous-time PDEs from sparse data with graph neural networks}, author={Valerii Iakovlev and Markus Heinonen and Harri L{\"a}hdesm{\"a}ki}, journal={ArXiv}, year={2021}, volume={abs/2006.08956} }

The behavior of many dynamical systems follow complex, yet still unknown partial differential equations (PDEs). While several machine learning methods have been proposed to learn PDEs directly from data, previous methods are limited to discrete-time approximations or make the limiting assumption of the observations arriving at regular grids. We propose a general continuous-time differential model for dynamical systems whose governing equations are parameterized by message passing graph neural…

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## References

SHOWING 1-10 OF 38 REFERENCES

Physics-aware Difference Graph Networks for Sparsely-Observed Dynamics

- Computer ScienceICLR
- 2020

A novel architecture named Physics-aware Difference Graph Networks (PA-DGN) is proposed that exploits neighboring information to learn finite differences inspired by physics equations and further leverages data-driven end-to-end learning to discover underlying dynamical relations between the spatial and temporal differences in given observations.

Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks

- Computer ScienceJ. Comput. Phys.
- 2020

DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data

- Computer ScienceArXiv
- 2019

The proposed DL-PDE method combines deep learning via neural networks and data-driven discovery of PDE via sparse regressions to discover the governing PDEs of underlying physical processes.

PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network

- Computer ScienceJ. Comput. Phys.
- 2019

PDE-Net: Learning PDEs from Data

- Computer ScienceICML
- 2018

Numerical experiments show that the PDE-Net has the potential to uncover the hidden PDE of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment.

Data driven approximation of parametrized PDEs by Reduced Basis and Neural Networks

- Computer ScienceJ. Comput. Phys.
- 2020

Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations

- Computer ScienceArXiv
- 2017

We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial…

Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations

- Computer ScienceArXiv
- 2017

This two part treatise introduces physics informed neural networks – neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations and demonstrates how these networks can be used to infer solutions topartial differential equations, and obtain physics-informed surrogate models that are fully differentiable with respect to all input coordinates and free parameters.

Solving parametric PDE problems with artificial neural networks

- Computer ScienceEuropean Journal of Applied Mathematics
- 2020

This work proposes using neural network to parameterise the physical quantity of interest as a function of input coefficients and demonstrates the simplicity and accuracy of the approach through notable examples of PDEs in engineering and physics.

DPM: A deep learning PDE augmentation method (with application to large-eddy simulation)

- Computer ScienceJ. Comput. Phys.
- 2020