# A two-stage physics-informed neural network method based on conserved quantities and applications in localized wave solutions

@inproceedings{Lin2021ATP, title={A two-stage physics-informed neural network method based on conserved quantities and applications in localized wave solutions}, author={Shuning Lin and Yong Chen}, year={2021} }

With the advantages of fast calculating speed and high precision, the physics-informed neural network method opens up a new approach for numerically solving nonlinear partial differential equations. Based on conserved quantities, we devise a two-stage PINN method which is tailored to the nature of equations by introducing features of physical systems into neural networks. Its remarkable advantage lies in that it can impose physical constraints from a global perspective. In stage one, the… Expand

#### Figures and Tables from this paper

#### 3 Citations

Modified physics-informed neural network method based on the conservation law constraint and its prediction of optical solitons

- Physics, Mathematics
- 2021

Based on conservation laws as one of the important integrable properties of nonlinear physical models, we design a modified physics-informed neural network method based on the conservation law… Expand

Neural Networks Enforcing Physical Symmetries in Nonlinear Dynamical Lattices: The Case Example of the Ablowitz-Ladik Model

- Physics
- 2021

In this work we introduce symmetry-preserving, physics-informed neural networks (S-PINNs) motivated by symmetries that are ubiquitous to solutions of nonlinear dynamical lattices. Although the use of… Expand

The data-driven vector localized waves of Manakov system using improved PINN approach

- Physics
- 2021

The study of vector localized waves, especially vector rogue waves (RWs), for the Manakov system has attracted more and more attention in many fields such as mathematical physics. An improved PINN… Expand

#### References

SHOWING 1-10 OF 79 REFERENCES

Solving localized wave solutions of the derivative nonlinear Schrodinger equation using an improved PINN method

- Physics
- 2021

The solving of the derivative nonlinear Schrodinger equation (DNLS) has attracted considerable attention in theoretical analysis and physical applications. Based on the physics-informed neural… Expand

Physics-informed neural networks for high-speed flows

- Physics
- 2020

Abstract In this work we investigate the possibility of using physics-informed neural networks (PINNs) to approximate the Euler equations that model high-speed aerodynamic flows. In particular, we… Expand

VPINNs: VARIATIONAL PHYSICS-INFORMED NEURAL NETWORKS FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS ̊

- 2019

Abstract. Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of… Expand

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

- Computer Science
- J. Comput. Phys.
- 2019

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

Localized waves of the coupled cubic–quintic nonlinear Schrödinger equations in nonlinear optics*

- Physics
- 2017

We investigate some novel localized waves on the plane wave background in the coupled cubic–quintic nonlinear Schrodinger (CCQNLS) equations through the generalized Darboux transformation (DT). A… Expand

fPINNs: Fractional Physics-Informed Neural Networks

- Physics, Computer Science
- SIAM J. Sci. Comput.
- 2019

This work extends PINNs to fractional PINNs (fPINNs) to solve space-time fractional advection-diffusion equations (fractional ADEs), and demonstrates their accuracy and effectiveness in solving multi-dimensional forward and inverse problems with forcing terms whose values are only known at randomly scattered spatio-temporal coordinates (black-box forcing terms). Expand

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

- Mathematics, Computer Science
- J. 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. Expand

Hidden physics models: Machine learning of nonlinear partial differential equations

- Computer Science, Mathematics
- J. Comput. Phys.
- 2018

Abstract While there is currently a lot of enthusiasm about “big data”, useful data is usually “small” and expensive to acquire. In this paper, we present a new paradigm of learning partial… Expand

Adaptive activation functions accelerate convergence in deep and physics-informed neural networks

- Computer Science, Physics
- J. Comput. Phys.
- 2020

It is theoretically proved that in the proposed method, gradient descent algorithms are not attracted to suboptimal critical points or local minima, and the proposed adaptive activation functions are shown to accelerate the minimization process of the loss values in standard deep learning benchmarks with and without data augmentation. Expand

Nonlocal symmetries, conservation laws and interaction solutions for the classical Boussinesq–Burgers equation

- Physics
- 2018

We consider the classical Boussinesq–Burgers (BB) equation, which describes the propagation of shallow water waves. Based on the truncated painlevé expansion method and consistent Riccati expansion… Expand