Deep interpolating wavelet network: for spatially adaptive fast discretization and feature capture guidance

Research on the adaptive multiscale interpolating wavelet transform has been focusing on the traditional collocation method and lacks attention to the computational efficiency and feature-capturing ability of the transform itself. We began by analyzing the essence of the adaptive multiscale interpolating wavelet transform. Subsequently, we developed a sparse-processed multiscale interpolating wavelet transform and introduced the deep interpolating wavelet network (DIWN). This transitioned from conventional numerical methods to deep learning methods, enhancing the computational speed and applicability of the adaptive multiscale interpolating wavelet transform. Simultaneously, we attempted to design DIWN as guidance for adaptive adjustment of physics-informed neural network (PINN) collocation points and designed deep interpolating wavelet physics-informed neural network. We analyzed the behavior of DIWN parameters and the impact of each attribute on architectural performance using simple two-dimensional signals. Furthermore, we validated the performance of DIWN in guiding the PINN optimization for solving the Bateman-Burgers’ equation, two-dimensional Poisson equation, Klein-Gordon equation, and Helmholtz equation. The experiments demonstrated that DIWN significantly improved the solution speed of the adaptive multiscale interpolating wavelet transform and also exhibited competitiveness comparable to mainstream adaptive methods in optimizing the PINN solution accuracy.