Solving the fully nonlinear Monge-Ampère equation using the Legendre-Kolmogorov-Arnold Network method
Journal:
arXiv
Published Date:
Apr 7, 2025
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-Amp\`ere 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-Amp\`ere 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.
