# Active training of physics-informed neural networks to aggregate and interpolate parametric solutions to the Navier-Stokes equations

@article{Arthurs2021ActiveTO, title={Active training of physics-informed neural networks to aggregate and interpolate parametric solutions to the Navier-Stokes equations}, author={Christopher J. Arthurs and Andrew P. King}, journal={Journal of Computational Physics}, year={2021}, volume={438}, pages={None - None} }

## 20 Citations

### On the Pareto Front of Physics-Informed Neural Networks

- Computer Science, PhysicsArXiv
- 2021

This paper sheds light on the training process of physics-informed neural networks, and uses the diffusion equation and Navier-Stokes equations in various test environments to analyze the effects of system parameters on the shape of the Pareto front.

### Design of Turing Systems with Physics-Informed Neural Networks

- Computer Science, Physics
- 2022

The utility of physics-informed neural networks for inverse parameter inference of reaction-diffusion systems to enhance scientiﬁc discovery and design is demonstrated.

### How PINNs cheat: Predicting chaotic motion of a double pendulum

- Physics
- 2022

Despite extensive research, physics-informed neural networks (PINNs) are still 1 difficult to train, especially when the optimization relies heavily on the physics 2 loss term. Convergence problems…

### A physics-informed learning approach to Bernoulli-type free boundary problems

- EducationComputers & Mathematics with Applications
- 2022

### Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations

- Computer Science
- 2022

In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs). We focus on a pair of benchmarks:…

### ViscoelasticNet: A physics informed neural network framework for stress discovery and model selection

- Engineering
- 2022

Viscoelastic ﬂuids are a class of ﬂuids that exhibit both viscous and elastic nature. Modeling such ﬂuids requires constitutive equations for the stress, and choosing the most appropriate…

### Improved Deep Neural Networks with Domain Decomposition in Solving Partial Differential Equations

- Computer ScienceJournal of Scientific Computing
- 2022

By investigations, it is shown that, although the neural networks structure and the loss function are complicated, the proposed method outperforms the classical PINNs with respect to training effectiveness, computational accuracy, and computational cost.

### On NeuroSymbolic Solutions for PDEs

- PsychologyArXiv
- 2022

To approximate complex functions with respect to Domain-splitting assisted approach on LaSalle-Bouchut inequality.

### The Optimal Error Estimate of the Fully Discrete Locally Stabilized Finite Volume Method for the Non-Stationary Navier-Stokes Problem

- Computer ScienceEntropy
- 2022

This paper proves the optimal estimations of a low-order spatial-temporal fully discrete method for the non-stationary Navier-Stokes Problem by adopting the traditional triangle P1−P0 trial function pair combined with macro element form to ensure local stability.

## References

SHOWING 1-10 OF 33 REFERENCES

### Physics-informed deep learning for incompressible laminar flows

- PhysicsTheoretical and Applied Mechanics Letters
- 2020

### Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data

- Computer Science
- 2020

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

- Computer ScienceJ. Comput. Phys.
- 2019

### Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data

- Computer ScienceArXiv
- 2018

HFM is a physics informed deep learning framework capable of encoding an important class of physical laws governing fluid motions, namely the Navier-Stokes equations, and can be used in physical and biomedical problems to extract valuable quantitative information for which direct measurements may not be possible.

### 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.

### Inferring solutions of differential equations using noisy multi-fidelity data

- Computer ScienceJ. Comput. Phys.
- 2017

### Physics-informed neural networks for solving nonlinear diffusivity and Biot’s equations

- PhysicsPloS one
- 2020

The methodology of physics-informed neural networks are investigated how to extend to solve both the forward and inverse problems in relation to the nonlinear diffusivity and Biot’s equations.

### Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems

- Computer ScienceJ. Comput. Phys.
- 2019

### 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…

### Accelerating Physics-Informed Neural Network Training with Prior Dictionaries

- Computer ScienceArXiv
- 2020

It is proved that under certain mild conditions, the prediction error made by neural networks can be bounded by expected loss of PDEs and boundary conditions and the error bounds applicable to PINNs and PD-PINNs are obtained.