Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism (2009.04544v5)

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.

• The paper introduces a novel SA-PINN model that uses self-adaptive soft attention weights to target difficult solution regions in stiff PDEs.
• It employs Gaussian Process regression to continuously map adaptive weights, enhancing training efficiency with stochastic gradient descent.
• Empirical results demonstrate reduced L2 error and fewer training epochs on benchmarks such as the Allen-Cahn and Helmholtz equations.

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

The paper presents a method that enhances the training of Physics-Informed Neural Networks (PINNs) through a novel self-adaptive mechanism inspired by soft attention methodologies. This method aims to address prevalent challenges in solving partial differential equations (PDEs), particularly the class of stiff PDEs, where traditional PINN approaches may suffer from convergence and accuracy issues.

The core innovation introduced in this work is the Self-Adaptive PINN (SA-PINN), which utilizes individual trainable weights applied to each training point. This mechanism allows the network to autonomously identify and focus on difficult regions of the solution space, thereby improving overall training efficiency and solution accuracy. The adaptation weights create a soft multiplicative attention mask, reminiscent of similar techniques employed in computer vision, which enhances the attention to less accurately predicted regions by increasing the weight as the corresponding loss increases.

A notable strength of this approach is the use of Gaussian Process regression to build a continuous map of self-adaptive weights, facilitating the application of stochastic gradient descent (SGD). This is particularly advantageous for problems where standard deterministic optimization falls short. The derivation of the Neural Tangent Kernel (NTK) for SA-PINNs provides theoretical insights into the training dynamics, specifically the effect of self-adaptive weights on the distribution of eigenvalues in the NTK matrix.

The empirical evaluation conducted on a variety of linear and nonlinear PDE benchmarks, including the Allen-Cahn, viscous Burgers, and Helmholtz equations, demonstrates that SA-PINNs outperform existing state-of-the-art PINNs with a lower L2 error and reduced training epochs. The numerical results confirm the algorithm's capability to handle stiff PDEs with high accuracy. Furthermore, the application of SA-PINNs to the wave equation highlights the benefits of SGD in this context, suggesting that future explorations into dynamic weighting strategies could yield deeper insights into training stability and accuracy.

The practical and theoretical implications of this research are substantial, particularly in areas where traditional numerical solvers face challenges, or where high-fidelity solutions are computationally expensive. The adaptability of SA-PINNs could streamline the modeling processes in engineering and physics applications, where complex boundary conditions and solution discontinuities are present.

Future work may explore further integration of dynamic weighting strategies, the development of new optimization algorithms tailored for PINNs, and deeper theoretical explorations into the relationship between PINNs and constrained optimization frameworks. This research contributes a significant advancement in the field of scientific machine learning, providing a robust tool for efficiently solving high-dimension and complex PDEs.