Solving the fully nonlinear Monge-Ampère equation using the Legendre-Kolmogorov-Arnold Network method

TL;DR

The paper introduces Legendre-KAN, a neural PDE solver for the fully nonlinear Monge-Ampère equation with Dirichlet boundary conditions, built on the Kolmogorov-Arnold representation and Legendre polynomials. It combines a KAN-based architecture with a Legendre basis, mapping inputs to the $[-1,1]$ domain and representing activations via Legendre expansions to improve accuracy and interpretability. An adaptive residual sampling strategy and a two-component interior/boundary loss within Adam optimization enable effective training, including high-dimensional and singular solutions, and are demonstrated on smooth, piecewise, and singular cases, as well as optimal transport-inspired image mappings. The results show strong convergence and accuracy advantages over conventional MLP approaches, with direct applicability to high-dimensional PDEs and practical geometric image transformations, underpinned by Brenier's optimal transport theory.

Abstract

In this paper, we propose a novel neural network framework, the Legendre-Kolmogorov-Arnold Network (Legendre-KAN) method, designed to solve fully nonlinear Monge-Ampère equations with Dirichlet boundary conditions. The architecture leverages the orthogonality of Legendre polynomials as basis functions, significantly enhancing both convergence speed and solution accuracy compared to traditional methods. Furthermore, the Kolmogorov-Arnold representation theorem provides a strong theoretical foundation for the interpretability and optimization of the network. We demonstrate the effectiveness of the proposed method through numerical examples, involving both smooth and singular solutions in various dimensions. This work not only addresses the challenges of solving high-dimensional and singular Monge-Ampère equations but also highlights the potential of neural network-based approaches for complex partial differential equations. Additionally, the method is applied to the optimal transport problem in image mapping, showcasing its practical utility in geometric image transformation. This approach is expected to pave the way for further enhancement of KAN-based applications and numerical solutions of PDEs across a wide range of scientific and engineering fields.

Figures (12)

• Figure 2.1: Legendre-KAN network
• Figure 3.1: Adaptive sampling for the examples
• Figure 4.1: MLP solution for problem \ref{['eq:23']}
• Figure 4.2: Legendre-KAN solution for problem \ref{['eq:23']}
• Figure 4.3: The comparison of the loss function for problem \ref{['eq:23']}