Render unto Numerics : Orthogonal Polynomial Neural Operator for PDEs with Non-periodic Boundary Conditions

@article{Liu2022RenderUN,
  title={Render unto Numerics : Orthogonal Polynomial Neural Operator for PDEs with Non-periodic Boundary Conditions},
  author={Ziyuan Liu and Haifeng Wang and Kaijun Bao and Xu Qian and Hong Zhang and Songhe Song},
  journal={ArXiv},
  year={2022},
  volume={abs/2206.12698}
}
By learning the map between function spaces using carefully designed deep neural networks, the operator learning become a focused field in recent several years, and have shown considerable efficiency over traditional numerical methods on solving complicated problems such as differential equations, but the method is still disturbed with the concern of its accuracy and reliability. In this paper, combined with the structures and technologies of a popular numerical method, i.e. the spectral method, a… 

Figures and Tables from this paper

References

SHOWING 1-10 OF 19 REFERENCES

Function-valued RKHS-based Operator Learning for Differential Equations

TLDR
This work incorporates function-valued reproducing kernel Hilbert spaces (function-valued RKHS) into their operator learning model, which shows that the approximate solution of target operator has a special form given by the representer theorem.

Model-Parallel Fourier Neural Operators as Learned Surrogates for Large-Scale Parametric PDEs

TLDR
This work proposes a model-parallel version of FNOs based on domain-decomposition of both the input data and network weights that is able to predict time-varying PDE solutions of over 3.2 billion vari-ables on Summit using up to 768 GPUs.

Multiwavelet-based Operator Learning for Differential Equations

TLDR
A multiwavelet-based neural operator learning scheme that compresses the associated operator’s kernel using fine-grained wavelets that achieves state-of-the-art in a range of datasets and exploits the fundamental properties of the operator's kernel which enable numerically efficient representation.

Wavelet neural operator: a neural operator for parametric partial differential equations

TLDR
A novel operator learning algorithm referred to as the Wavelet Neural Operator (WNO) that blends integral kernel with wavelet transformation and is used to build a digital twin capable of predicting Earth’s air temperature based on available historical data.

On universal approximation and error bounds for Fourier Neural Operators

TLDR
It is shown that the size of the FNO, approximating operators associated with a Darcy type elliptic PDE and with the incompressible Navier-Stokes equations of fluid dynamics, only increases sub (log)-linearly in terms of the reciprocal of the error.

Fourier Neural Operator for Solving Subsurface Oil/Water Two-Phase Flow Partial Differential Equation

TLDR
A deep-learning-based model is developed to solve three categories of problems controlled by the subsurface 2D oil/water two-phase flow PDE based on the FNO, a recently proposed high-efficiency PDE solution architecture that overcomes the shortcomings of popular algorithms such as physics-informed neural networks and fully convolutional network.

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

TLDR
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.

Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions

TLDR
The numerical results clearly show that the Deep Nitsche Method is naturally nonlinear, naturally adaptive and has the potential to work on rather high dimensions.