Numerical and deep learning methods for diffusion model with fractional Laplacian operator and its application to signal denoising

In modern signal processing applications, denoising signals remain an essential task, as it is difficult to reduce noise without compromising essential structural information. While standard Laplacian operators are limited in their capacity to handle long-range interactions and maintain fine-scale features, classical diffusion models offer a sound mathematical foundation for signal smoothing. To overcome these limitations, in this study, we developed a novel numerical and deep learning approach driven by a nonlocal fractional Laplacian operator of the form $(-\mathrm{\Delta }{)}^s$ $(0<s<1)$, which captures long-range interactions within signals. To accurately approximate the fractional Laplacian dynamics, a classical second-order finite difference (FD) discretization of the Laplacian was constructed in space, and the fractional operator was defined through its spectral matrix power. The resulting semi-discrete system was integrated in time using an unconditionally stable Crank-Nicolson (CN) scheme. In addition to this model-based method, a convolutional neural network (CNN) architecture was employed as a refinement step, trained to learn the nonlinear mapping from noisy to clean signals. To evaluate the denoising capability and effectiveness of the proposed methods, different input signals contaminated with additive white Gaussian noise at prescribed signal-to-noise (SNR) ratios were considered. The SNR and root mean square error (RMSE) were used as quantitative performance measures. The results showed that the FD-CN and the CNN methods perform well in conjunction. The numerical scheme smooths the data in a numerically controlled way, and the CNN captures complex local structures and adapts to non-Gaussian noise patterns. In addition, the proposed framework was extended to multichannel real-world ECG signals, demonstrating its robustness in handling correlated noise across multiple leads. The multichannel experiments confirmed that the FD-CN and CNN-based approaches remain effective under realistic, real-data conditions.