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 that may involve limited data availability. However, solving differential equations with rapid oscillations, steep gradients, or singular behavior becomes a challenge 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 in wavelet space using a family of 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 the training process to search for solutions within the wavelet domain, where the multiscale characteristics are less pronounced compared to the physical domain. This facilitates more efficient training for this class of problems. Furthermore, the proposed model does not rely on automatic differentiation (AD) for derivatives involved in the loss function and does not require any prior information regarding the behavior of the solution, such as the location of abrupt features. The removal of the AD requirement significantly reduces training time while maintaining accuracy. Thus, through a strategic fusion of wavelets with PINNs, W-PINNs excel at capturing localized non-linear information, making them well-suited for problems showing abrupt behavior in certain regions, such as singularly perturbed and other multiscale problems. We further analyze the convergence behavior of W-PINN through a comparative study using the Neural Tangent Kernel (NTK) theory. The efficiency and accuracy of the proposed neural network model are demonstrated in various problems, i.e., the FitzHugh-Nagumo (FHN) model, the Helmholtz equation, the Maxwell equation, the Allen-Cahn equation, and lid-driven cavity flow, along with other highly singularly perturbed non-linear differential equations.