Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism

• URL: http://arxiv.org/abs/2009.04544v5
• Date: Tue, 18 Jun 2024 23:09:32 GMT
• Title: Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism
• Authors: Levi McClenny, Ulisses Braga-Neto,
• Abstract summary: Physics-Informed Neural Networks (PINNs) have emerged as a promising application of deep neural networks to the numerical solution of nonlinear partial differential equations (PDEs) We propose a fundamentally new way to train PINNs adaptively, where the adaptation weights are fully trainable and applied to each training point individually. In numerical experiments with several linear and nonlinear benchmark problems, the SA-PINN outperformed other state-of-the-art PINN algorithm in L2 error.
• Score: 1.6114012813668932
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Physics-Informed Neural Networks (PINNs) have emerged recently as a promising application of deep neural networks to the numerical solution of nonlinear partial differential equations (PDEs). However, it has been recognized that adaptive procedures are needed to force the neural network to fit accurately the stubborn spots in the solution of "stiff" PDEs. In this paper, we propose a fundamentally new way to train PINNs adaptively, where the adaptation weights are fully trainable and applied to each training point individually, so the neural network learns autonomously which regions of the solution are difficult and is forced to focus on them. The self-adaptation weights specify a soft multiplicative soft attention mask, which is reminiscent of similar mechanisms used in computer vision. The basic idea behind these SA-PINNs is to make the weights increase as the corresponding losses increase, which is accomplished by training the network to simultaneously minimize the losses and maximize the weights. In addition, we show how to build a continuous map of self-adaptive weights using Gaussian Process regression, which allows the use of stochastic gradient descent in problems where conventional gradient descent is not enough to produce accurate solutions. Finally, we derive the Neural Tangent Kernel matrix for SA-PINNs and use it to obtain a heuristic understanding of the effect of the self-adaptive weights on the dynamics of training in the limiting case of infinitely-wide PINNs, which suggests that SA-PINNs work by producing a smooth equalization of the eigenvalues of the NTK matrix corresponding to the different loss terms. In numerical experiments with several linear and nonlinear benchmark problems, the SA-PINN outperformed other state-of-the-art PINN algorithm in L2 error, while using a smaller number of training epochs.

Related papers

• HyResPINNs: Adaptive Hybrid Residual Networks for Learning Optimal Combinations of Neural and RBF Components for Physics-Informed Modeling [22.689531776611084]
  We present a new class of PINNs called HyResPINNs, which augment traditional PINNs with adaptive hybrid residual blocks. A key feature of our method is the inclusion of adaptive combination parameters within each residual block. We show that HyResPINNs are more robust to training point locations and neural network architectures than traditional PINNs.
  arXiv  Detail & Related papers  (2024-10-04T16:21:14Z)
• Enriched Physics-informed Neural Networks for Dynamic Poisson-Nernst-Planck Systems [0.8192907805418583]
  This paper proposes a meshless deep learning algorithm, enriched physics-informed neural networks (EPINNs) to solve dynamic Poisson-Nernst-Planck (PNP) equations. The EPINNs takes the traditional physics-informed neural networks as the foundation framework, and adds the adaptive loss weight to balance the loss functions. Numerical results indicate that the new method has better applicability than traditional numerical methods in solving such coupled nonlinear systems.
  arXiv  Detail & Related papers  (2024-02-01T02:57:07Z)
• Collocation-based Robust Variational Physics-Informed Neural Networks (CRVPINN) [0.0]
  Physics-Informed Neural Networks (PINNs) have been successfully applied to solve Partial Differential Equations (PDEs) Recent work of Robust Variational Physics-Informed Neural Networks (RVPINNs) highlights the importance of conveniently translating the norms of the underlying continuum-level spaces to the discrete level. In this work, we accelerate the implementation of RVPINN, establishing a LU factorization of sparse Gram matrix in a kind of point-collocation scheme with the same spirit as original PINNs.
  arXiv  Detail & Related papers  (2024-01-04T14:42:29Z)
• Fixing the NTK: From Neural Network Linearizations to Exact Convex Programs [63.768739279562105]
  We show that for a particular choice of mask weights that do not depend on the learning targets, this kernel is equivalent to the NTK of the gated ReLU network on the training data. A consequence of this lack of dependence on the targets is that the NTK cannot perform better than the optimal MKL kernel on the training set.
  arXiv  Detail & Related papers  (2023-09-26T17:42:52Z)
• GPT-PINN: Generative Pre-Trained Physics-Informed Neural Networks toward non-intrusive Meta-learning of parametric PDEs [0.0]
  We propose the Generative Pre-Trained PINN (GPT-PINN) to mitigate both challenges in the setting of parametric PDEs. As a network of networks, its outer-/meta-network is hyper-reduced with only one hidden layer having significantly reduced number of neurons. The meta-network adaptively learns'' the parametric dependence of the system and grows'' this hidden layer one neuron at a time.
  arXiv  Detail & Related papers  (2023-03-27T02:22:09Z)
• Globally Optimal Training of Neural Networks with Threshold Activation Functions [63.03759813952481]
  We study weight decay regularized training problems of deep neural networks with threshold activations. We derive a simplified convex optimization formulation when the dataset can be shattered at a certain layer of the network.
  arXiv  Detail & Related papers  (2023-03-06T18:59:13Z)
• 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)
• Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs) [1.7403133838762446]
  Physics-Informed Neural Networks (PINNs) are powerful approximators of solutions to nonlinear partial differential equations (PDEs) We propose an Augmented Lagrangian relaxation method for PINNs (AL-PINNs) We demonstrate through various numerical experiments that AL-PINNs yield a much smaller relative error compared with that of state-of-the-art adaptive loss-balancing algorithms.
  arXiv  Detail & Related papers  (2022-04-29T08:33:11Z)
• 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)
• LocalDrop: A Hybrid Regularization for Deep Neural Networks [98.30782118441158]
  We propose a new approach for the regularization of neural networks by the local Rademacher complexity called LocalDrop. A new regularization function for both fully-connected networks (FCNs) and convolutional neural networks (CNNs) has been developed based on the proposed upper bound of the local Rademacher complexity.
  arXiv  Detail & Related papers  (2021-03-01T03:10:11Z)
• Revisiting Initialization of Neural Networks [72.24615341588846]
  We propose a rigorous estimation of the global curvature of weights across layers by approximating and controlling the norm of their Hessian matrix. Our experiments on Word2Vec and the MNIST/CIFAR image classification tasks confirm that tracking the Hessian norm is a useful diagnostic tool.
  arXiv  Detail & Related papers  (2020-04-20T18:12:56Z)

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.