Aleksei Krasnikov | Searcharxiv

arXiv2026

Learning Ecological and Epidemic Processes using Neural ODEs, Kolmogorov-Arnold Network ODEs and SINDy

We consider epidemic and ecological models to investigate their coupled dynamics. Starting with the classical Susceptible-Infected-Recovered (SIR) model for basic epidemic behavior and the predator-prey (Lotka-Volterra, LV) system for ecological interactions, we then combine these frameworks into a coupled Lotka-Volterra-Susceptible-Infected-Susceptible (LVSIS) model. The resulting system consists of four differential equations describing the evolution of susceptible and infected prey and predator populations, incorporating ecological interactions, disease transmission, and spatial dispersal. To learn the underlying dynamics directly from data, we employ several data-driven modeling frameworks: Neural Ordinary Differential Equations (Neural ODEs), Kolmogorov-Arnold Network Ordinary Differential Equations (KANODEs), and Sparse Identification of Nonlinear Dynamics (SINDy). Numerical experiments based on synthetic data are conducted to investigate the learning ability of these models in capturing the epidemic and ecological behavior. We further extend our approach to spatio-temporal models, aiming to uncover hidden local couplings.

math.NA↗

arXiv2024

Multiscale method for image denoising using nonlinear diffusion process: local denoising and spectral multiscale basis functions

We consider image denoising using a nonlinear diffusion process, where we solve unsteady partial differential equations with nonlinear coefficients. The noised image is given as an initial condition, and nonlinear coefficients are used to preserve the main image features. In this paper, we present a multiscale method for the resulting nonlinear parabolic equation in order to construct an efficient solver. To both filter out noise and preserve essential image features during the denoising process, we utilize a time-dependent nonlinear diffusion model. Here, the noised image is fed as an initial condition and the denoised image is stimulated with given parameters. We numerically implement this model by constructing a discrete system for a given image resolution using a finite volume method and employing an implicit time approximation scheme to avoid time-step restriction. However, the resulting discrete system size is proportional to the number of pixels which leads to computationally expensive numerical algorithms for high-resolution images. In order to reduce the size of the system and construct efficient computational algorithms, we construct a coarse-resolution representation of the system. We incorporate local noise reduction in the coarsening process to construct an efficient algorithm with fewer denoising iterations. We propose a computational approach with two main ingredients: (1) performing local image denoising in each local domain of basis support; and (2) constructing multiscale basis functions to construct a coarse resolution representation by a Galerkin coupling. We present numerical results for several classic and high-resolution image datasets to demonstrate the effectiveness of the proposed multiscale approach with local denoising and local multiscale representation.

math.NA↗