# Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems

@article{Tanyu2022DeepLM, title={Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems}, author={Derick Nganyu Tanyu and Jianfeng Ning and Tom Freudenberg and Nick Heilenk{\"o}tter and Andreas Rademacher and Uwe Iben and Peter Maass}, journal={ArXiv}, year={2022}, volume={abs/2212.03130} }

Recent years have witnessed a growth in mathematics for deep learning—which seeks a deeper understanding of the concepts of deep learning with mathematics, and explores how to make it more robust—and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network architectures…

## 3 Citations

### Can Physics-Informed Neural Networks beat the Finite Element Method?

- Computer Science, PhysicsArXiv
- 2023

In terms of solution time and accuracy, physics-informed neural networks have not been able to outperform the finite element method in this study, but in some experiments, they were faster at evaluating the solved PDE.

### Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients

- Computer ScienceArXiv
- 2022

The Ehrenpreis-Palamodov fundamental principle is applied, which works like a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs, and can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise deﬁned initial and boundary conditions.

### Hybrid Neural-Network FEM Approximation of Diffusion Coefficient in Elliptic and Parabolic Problems

- Computer Science, MathematicsArXiv
- 2023

This work derives \textsl{a priori} error estimates in the standard $L^2(\Omega)$ norm for the numerical reconstruction, under a positivity condition which can be verified for a large class of problem data.

## References

SHOWING 1-10 OF 113 REFERENCES

### DeepXDE: A Deep Learning Library for Solving Differential Equations

- Computer ScienceAAAI Spring Symposium: MLPS
- 2020

An overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation, and a new residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs.

### Deep Neural Networks Motivated by Partial Differential Equations

- Computer ScienceJournal of Mathematical Imaging and Vision
- 2019

A new PDE interpretation of a class of deep convolutional neural networks (CNN) that are commonly used to learn from speech, image, and video data is established and three new ResNet architectures are derived that fall into two new classes: parabolic and hyperbolic CNNs.

### Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning

- Computer ScienceNonlinearity
- 2021

It is demonstrated to the reader that studying PDEs as well as control and variational problems in very high dimensions might very well be among the most promising new directions in mathematics and scientific computing in the near future.

### Stable architectures for deep neural networks

- Computer ScienceArXiv
- 2017

This paper relates the exploding and vanishing gradient phenomenon to the stability of the discrete ODE and presents several strategies for stabilizing deep learning for very deep networks.

### Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations

- Computer ScienceJ. Mach. Learn. Res.
- 2018

This work puts forth a deep learning approach for discovering nonlinear partial differential equations from scattered and potentially noisy observations in space and time by approximate the unknown solution as well as the nonlinear dynamics by two deep neural networks.

### Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

- Computer ScienceJ. Comput. Phys.
- 2019

### Improved Architectures and Training Algorithms for Deep Operator Networks

- Computer ScienceJournal of Scientific Computing
- 2022

This work analyzes the training dynamics of deep operator networks (DeepONets) through the lens of Neural Tangent Kernel theory, and reveals a bias that favors the approximation of functions with larger magnitudes.

### Enhanced DeepONet for Modeling Partial Differential Operators Considering Multiple Input Functions

- Computer ScienceArXiv
- 2022

New Enhanced DeepONet or EDeepONet high-level neural network structure is proposed, in which two input functions are represented by two branch DNN sub-networks, which are then connected with output truck network via inner product to generate the output of the whole neural network.

### DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators

- Computer ScienceArXiv
- 2019

This work proposes deep operator networks (DeepONets) to learn operators accurately and efficiently from a relatively small dataset, and demonstrates that DeepONet significantly reduces the generalization error compared to the fully-connected networks.

### Fourier Neural Operator for Parametric Partial Differential Equations

- Computer ScienceICLR
- 2021

This work forms a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture and shows state-of-the-art performance compared to existing neural network methodologies.