Methods for the Numerical Analysis of Boundary Value Problem of Partial Differential Equations Based on Kolmogorov Superposition Theorem

TL;DR

This paper presents a novel numerical method leveraging the Kolmogorov superposition theorem to convert partial differential equations into systems of ordinary differential equations, demonstrated on the Poisson equation.

Contribution

It introduces a new approach for transforming PDEs into ODE systems using Kolmogorov's theorem, enabling alternative numerical solutions.

Findings

Successfully applied to the Poisson equation

Generated ODE systems match PDE solutions

Demonstrated feasibility of the method

Abstract

This research introduces a new method for the transition from partial to ordinary differential equations that is based on the Kolmogorov superposition theorem. In this paper, we discuss the numerical implementation of the Kolmogorov theorem and propose an approach that allows us to apply the theorem to represent partial derivatives of multivariate function as a combination of ordinary derivatives of univariate functions. We tested the method by running a numerical experiment with the Poisson equation. As a result, we managed to get a system of ordinary differential equations whose solution coincides with a solution of the initial partial differential equation.