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

Abstract: Physics-informed neural networks (PINNs) are a class of deep learning models that utilize physics in the form of differential equations to address complex problems, including ones that may involve limited data availability. However, tackling solutions of differential equations with rapid oscillations, steep gradients, or singular behavior becomes challenging for PINNs. Considering these challenges, we designed an efficient wavelet-based PINNs (W-PINNs) model to address this class of 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 the dynamics of complex physical phenomena. The architecture allows the training process to search for a solution within the wavelet space, making the process faster and more accurate. Further, 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 and multiscale problems. The efficiency and accuracy of the proposed neural network model are demonstrated in various 1D and 2D test problems, i.e., the FitzHugh-Nagumo (FHN) model, the Helmholtz equation, the Maxwell's equation, lid-driven cavity flow, and the Allen-Cahn equation, along with other highly singularly perturbed nonlinear differential equations. The proposed model significantly improves with traditional PINNs, recently developed wavelet-based PINNs, and other state-of-the-art methods.