LowNet: Looped Wavelet Neural Networks with 24 Learnable Parameters for Solving Poisson Equations

Poisson equations frequently appear in various fields, such as mathematics, physics and informatics. Solving Poisson equation, however, is challenging. Traditional methods are not data-specific, although have high certainty. They require a large number of iterations due to the CFL numerical stability. In contrast, deep learning methods are data-driven and more efficient, but require accurate training data and suffer from heavy computation. To tackle these issues, we propose a wavelet-inspired Poisson solver with 24 trainable scalars. The proposed network has several advantages. First, it is compact. It has four independent 5 times5 kernels, each represented by six parameters, leading to 24 scalars in total. Such compact format can be deployed on most devices. Second, thanks to the loop design, it achieves higher accuracy than previous methods. A residual loop reuses the wavelet block without increasing its parameters. Numerical experiments confirm that such loop indeed improves the solving accuracy. On 10000 images, one, two, and three loops of the wavelet block lead to PSNRs of 47.27, 74.14, and 100.74 dB, respectively. Third, the proposed method is computationally fast. Thanks to the wavelet transform, the proposed method has linear computation complexity, leading to high computation performance. In short, we propose a novel looped wavelet neural network with compactness, accuracy and efficiency for solving Poisson equations. Our method can be applied in various fields, such as computational fluid dynamics, image reconstruction, biomedical engineering, electromagnetic field simulation.