Gradient-enhanced physics-informed neural networks for forward and
inverse PDE problems
- URL: http://arxiv.org/abs/2111.02801v1
- Date: Mon, 1 Nov 2021 18:01:38 GMT
- Title: Gradient-enhanced physics-informed neural networks for forward and
inverse PDE problems
- Authors: Jeremy Yu, Lu Lu, Xuhui Meng, George Em Karniadakis
- Abstract summary: Deep learning has been shown to be an effective tool in solving partial differential equations (PDEs) through physics-informed neural networks (PINNs)
PINNs embed the PDE residual into the loss function of the neural network, and have been successfully employed to solve diverse forward and inverse PDE problems.
Here, we propose a new method, gradient-enhanced physics-informed neural networks (gPINNs) for improving the accuracy and training efficiency of PINNs.
- Score: 2.0062792633909026
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Deep learning has been shown to be an effective tool in solving partial
differential equations (PDEs) through physics-informed neural networks (PINNs).
PINNs embed the PDE residual into the loss function of the neural network, and
have been successfully employed to solve diverse forward and inverse PDE
problems. However, one disadvantage of the first generation of PINNs is that
they usually have limited accuracy even with many training points. Here, we
propose a new method, gradient-enhanced physics-informed neural networks
(gPINNs), for improving the accuracy and training efficiency of PINNs. gPINNs
leverage gradient information of the PDE residual and embed the gradient into
the loss function. We tested gPINNs extensively and demonstrated the
effectiveness of gPINNs in both forward and inverse PDE problems. Our numerical
results show that gPINN performs better than PINN with fewer training points.
Furthermore, we combined gPINN with the method of residual-based adaptive
refinement (RAR), a method for improving the distribution of training points
adaptively during training, to further improve the performance of gPINN,
especially in PDEs with solutions that have steep gradients.
Related papers
- VS-PINN: A fast and efficient training of physics-informed neural networks using variable-scaling methods for solving PDEs with stiff behavior [0.0]
We propose a new method for training PINNs using variable-scaling techniques.
We will demonstrate the effectiveness of the proposed method for these problems and confirm that it can significantly improve the training efficiency and performance of PINNs.
arXiv Detail & Related papers (2024-06-10T14:11:15Z) - RoPINN: Region Optimized Physics-Informed Neural Networks [66.38369833561039]
Physics-informed neural networks (PINNs) have been widely applied to solve partial differential equations (PDEs)
This paper proposes and theoretically studies a new training paradigm as region optimization.
A practical training algorithm, Region Optimized PINN (RoPINN), is seamlessly derived from this new paradigm.
arXiv Detail & Related papers (2024-05-23T09:45:57Z) - Implicit Stochastic Gradient Descent for Training Physics-informed
Neural Networks [51.92362217307946]
Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems.
PINNs are trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features.
In this paper, we propose to employ implicit gradient descent (ISGD) method to train PINNs for improving the stability of training process.
arXiv Detail & Related papers (2023-03-03T08:17:47Z) - SVD-PINNs: Transfer Learning of Physics-Informed Neural Networks via Singular Value Decomposition [24.422082821785487]
One neural network corresponds to one partial differential equations.
In practice, we usually need to solve a class of PDEs, not just one.
We propose a transfer learning method of PINNs via keeping singular vectors and optimizing singular values.
arXiv Detail & Related papers (2022-11-16T08:46:10Z) - Auto-PINN: Understanding and Optimizing Physics-Informed Neural
Architecture [77.59766598165551]
Physics-informed neural networks (PINNs) are revolutionizing science and engineering practice by bringing together the power of deep learning to bear on scientific computation.
Here, we propose Auto-PINN, which employs Neural Architecture Search (NAS) techniques to PINN design.
A comprehensive set of pre-experiments using standard PDE benchmarks allows us to probe the structure-performance relationship in PINNs.
arXiv Detail & Related papers (2022-05-27T03:24:31Z) - Revisiting PINNs: Generative Adversarial Physics-informed Neural
Networks and Point-weighting Method [70.19159220248805]
Physics-informed neural networks (PINNs) provide a deep learning framework for numerically solving partial differential equations (PDEs)
We propose the generative adversarial neural network (GA-PINN), which integrates the generative adversarial (GA) mechanism with the structure of PINNs.
Inspired from the weighting strategy of the Adaboost method, we then introduce a point-weighting (PW) method to improve the training efficiency of PINNs.
arXiv Detail & Related papers (2022-05-18T06:50:44Z) - Learning Physics-Informed Neural Networks without Stacked
Back-propagation [82.26566759276105]
We develop a novel approach that can significantly accelerate the training of Physics-Informed Neural Networks.
In particular, we parameterize the PDE solution by the Gaussian smoothed model and show that, derived from Stein's Identity, the second-order derivatives can be efficiently calculated without back-propagation.
Experimental results show that our proposed method can achieve competitive error compared to standard PINN training but is two orders of magnitude faster.
arXiv Detail & Related papers (2022-02-18T18:07:54Z) - PSO-PINN: Physics-Informed Neural Networks Trained with Particle Swarm
Optimization [0.0]
We propose the use of a hybrid particle swarm optimization and gradient descent approach to train PINNs.
The resulting PSO-PINN algorithm mitigates the undesired behaviors of PINNs trained with standard gradient descent.
Experimental results show that PSO-PINN consistently outperforms a baseline PINN trained with Adam gradient descent.
arXiv Detail & Related papers (2022-02-04T02:21:31Z) - Physics-Informed Neural Operator for Learning Partial Differential
Equations [55.406540167010014]
PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator.
The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
arXiv Detail & Related papers (2021-11-06T03:41:34Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.