标题： 利用Legendre-Kolmogorov-Arnold网络方法求解完全非线性Monge-Ampère方程 显示英文标题

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

摘要： 本文提出了一种新颖的神经网络框架——勒让德-柯尔莫戈洛夫-阿诺德网络（Legendre-KAN）方法，旨在求解具有狄利克雷边界条件的完全非线性蒙日-安培方程。该架构利用了勒让德多项式作为基函数的正交性，与传统方法相比显著提高了收敛速度和解的准确性。此外，柯尔莫戈洛夫-阿诺德表示定理为网络的可解释性和优化提供了坚实的理论基础。我们通过数值例子展示了所提出方法的有效性，这些例子涉及各种维度下的光滑和奇异解。这项工作不仅解决了求解高维和奇异蒙日-安培方程的挑战，还强调了基于神经网络的方法在复杂偏微分方程中的潜力。此外，该方法被应用于图像映射中的最优传输问题，展示了其在几何图像变换中的实际效用。此方法有望为进一步增强基于KAN的应用以及科学和工程领域广泛偏微分方程的数值解铺平道路。 显示英文摘要

摘要： 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.