• Corpus ID: 243986031

Solving time-dependent parametric PDEs by multiclass classification-based reduced order model

@article{Cui2021SolvingTP,
  title={Solving time-dependent parametric PDEs by multiclass classification-based reduced order model},
  author={Chen Cui and Kai Jiang and Shi Shu},
  journal={ArXiv},
  year={2021},
  volume={abs/2111.06078}
}
In this paper, we propose a network model, the multiclass classification-based reduced order model (MC-ROM), for solving time-dependent parametric partial differential equations (PPDEs). This work is inspired by the observation of applying the deep learning-based reduced order model (DL-ROM) [14] to solve diffusion-dominant PPDEs. We find that the DL-ROM has a good approximation for some special model parameters, but it cannot approximate the drastic changes of the solution as time evolves. Based… 

References

SHOWING 1-10 OF 45 REFERENCES

A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs

Numerical results indicate that DL-ROMs whose dimension is equal to the intrinsic dimensionality of the PDE solutions manifold are able to efficiently approximate the solution of parametrized PDEs, especially in cases for which a huge number of POD modes would have been necessary to achieve the same degree of accuracy.

Support vector machines for multi-class pattern recognition

A formulation of the SVM is proposed that enables a multi-class pattern recognition problem to be solved in a single optimisation and a similar generalization of linear programming machines is proposed.

Learning the solution operator of parametric partial differential equations with physics-informed DeepONets

This work proposes a novel model class coined as physics-informed DeepONets, which introduces an effective regularization mechanism for biasing the outputs of DeepOnet models towards ensuring physical consistency, and demonstrates the effectiveness of the proposed framework through a series of comprehensive numerical studies across various types of PDEs.

Physics-informed machine learning for reduced-order modeling of nonlinear problems

Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEs

Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning

It is shown that a single deep neural network trained on simulated data is capable of learning the solution functions of an entire family of PDEs on a full space-time region and that the proposed method does not suffer from the curse of dimensionality, distinguishing it from almost all standard numerical methods for P DEs.

Fourier Neural Operator for Parametric Partial Differential Equations

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.

An autoencoder‐based reduced‐order model for eigenvalue problems with application to neutron diffusion

Using an autoencoder for dimensionality reduction, this article presents a novel projection‐based reduced‐order model for eigenvalue problems, compared with the standard POD‐Galerkin approach and applied to two test cases taken from the field of nuclear reactor physics.

Model Reduction and Neural Networks for Parametric PDEs

A neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation is developed.

A long short-term memory embedding for hybrid uplifted reduced order models