An efficient wavelet-based physics-informed neural networks for singularly perturbed problems

Himanshu Pandey, Anshima Singh, Ratikanta Behera
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Abstract

Physics-informed neural networks (PINNs) are a class of deep learning models that utilize physics as differential equations to address complex problems, including ones that may involve limited data availability. However, tackling solutions of differential equations with oscillations or singular perturbations and shock-like structures becomes challenging for PINNs. Considering these challenges, we designed an efficient wavelet-based PINNs (W-PINNs) model to solve singularly perturbed differential equations. Here, we represent the solution in wavelet space using a family of smooth-compactly supported wavelets. This framework represents the solution of a differential equation with significantly fewer degrees of freedom while still retaining in capturing, identifying, and analyzing the local structure of complex physical phenomena. The architecture allows the training process to search for a solution within wavelet space, making the process faster and more accurate. The proposed model does not rely on automatic differentiations for derivatives involved in differential equations and does not require any prior information regarding the behavior of the solution, such as the location of abrupt features. Thus, through a strategic fusion of wavelets with PINNs, W-PINNs excel at capturing localized nonlinear information, making them well-suited for problems showing abrupt behavior in certain regions, such as singularly perturbed problems. The efficiency and accuracy of the proposed neural network model are demonstrated in various test problems, i.e., highly singularly perturbed nonlinear differential equations, the FitzHugh-Nagumo (FHN), and Predator-prey interaction models. The proposed design model exhibits impressive comparisons with traditional PINNs and the recently developed wavelet-based PINNs, which use wavelets as an activation function for solving nonlinear differential equations.
用于奇异扰动问题的高效小波物理信息神经网络
物理信息神经网络(PINNs)是一类利用物理微分方程解决复杂问题的深度学习模型,包括那些可能涉及有限数据可用性的问题。然而,对于 PINNs 来说,处理具有振荡或奇异扰动和类似冲击结构的微分方程求解变得具有挑战性。考虑到这些挑战,我们设计了一种高效的基于小波的 PINNs(W-PINNs)模型来求解奇异扰动微分方程。在这里,我们使用平滑-紧凑支持的小波系列来表示小波空间中的解。这种框架表示微分方程的解,自由度大大降低,同时仍能捕捉、识别和分析复杂物理现象的局部结构。这种架构允许训练过程在小波空间内搜索解,从而使过程更快、更准确。所提出的模型不依赖于微分方程导数的自动微分,也不需要任何有关解的行为的先验信息,如突变特征的位置。因此,通过小波与 PINNs 的策略性融合,W-PINNs 擅长捕捉局部非线性信息,使其非常适合处理在某些区域显示中断行为的问题,如奇异扰动问题。在各种测试问题中,即高度奇异扰动非线性微分方程、FitzHugh-Nagumo(FHN)和捕食者-猎物相互作用模型中,证明了所提出的神经网络模型的高效性和准确性。所提出的设计模型与传统 PINN 和最近开发的基于小波的 PINN(使用小波作为激活函数求解非线性微分方程)进行了比较,结果令人印象深刻。
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