An efficient wavelet-based physics-informed neural network for multiscale problems

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 those with limited data availability. However, solving differential equations with rapid oscillations, steep gradients, or singular behavior remains challenging for PINNs. To address this, we propose an efficient wavelet-based physics-informed neural network (W-PINN) that learns solutions in wavelet space. Here, we represent the solution using localized wavelets. This framework represents the solution of a differential equation with significantly fewer degrees of freedom while retaining the dynamics of complex physical phenomena. The proposed architecture enables training to search for solutions within the wavelet domain, where multiscale characteristics are less pronounced compared to the physical domain. This facilitates more efficient training for such problems. Furthermore, the proposed model does not rely on automatic differentiation for derivatives in the loss function and does not require prior information regarding the behavior of the solution, such as the location of abrupt features. The removal of AD significantly reduces training time while maintaining accuracy. Thus, through a strategic fusion of wavelets with PINNs, W-PINNs capture localized nonlinear information, making them well-suited for problems with abrupt behavior, such as singularly perturbed and other multiscale problems. We further analyze the convergence behavior of W-PINN through a comparative study using Neural Tangent Kernel theory. The efficiency and accuracy of the proposed model are demonstrated across various problems, including the FitzHugh--Nagumo (FHN) model, Helmholtz equation, Maxwell equation, Allen--Cahn equation, and lid-driven cavity flow, along with other highly singularly perturbed nonlinear differential equations.