Wavelet-based enrichment for physics informed neural networks to approximate localized and heterogeneous solutions in solid mechanics

Recent research has highlighted the potential of physics-informed neural networks (PINNs) as an efficient methodology for approximating solutions of boundary value problems in solid mechanics. Nevertheless, their ability to accurately capture highly heterogeneous solutions of complex problems remains limited and requires further investigation. The present paper explores new strategies to address this challenge. In line with existing approaches based on local refinement of collocation (training) points, a weighted version of the loss function is first proposed to better balance the physical residuals across the entire computational domain. Although this modification improves overall performance, the approximation accuracy remains unsatisfactory. To overcome this limitation, an enriched version of PINN is developed to more effectively capture locally heterogeneous distributions of state variables. Specifically, wavelet-based enrichment functions are designed to approximate local high-frequency components of the full-field solution, thereby simplifying the task of the neural network, which is then required only to approximate the global smooth component of the solution. This approach achieves satisfactory accuracy even with relatively simple neural network architectures and few collocation points, as demonstrated through several benchmark problems. Therefore, the proposed enrichment concept represents a promising direction for further improving the performance of PINNs as solvers in computational mechanics, paving the way for their application to more complex problems.