标题： 基于拉普拉斯特征函数的神经算子用于学习非线性偏微分方程 显示英文标题

标题： Laplacian Eigenfunction-Based Neural Operator for Learning Nonlinear Partial Differential Equations

摘要： 学习非线性偏微分方程（PDEs）正在许多科学和工程学科中兴起，推动了流体力学、材料科学和生物系统等领域的发展。 在这项工作中，我们介绍了基于拉普拉斯特征函数的神经算子（LE-NO），这是一种高效学习PDEs中非线性项的框架，特别关注非线性抛物型方程。 通过利用数据驱动的方法来建模右侧的非线性算子，LE-NO 框架使用拉普拉斯特征函数作为基函数，从而能够有效地逼近非线性项。 这种方法通过直接计算逆拉普拉斯矩阵降低了问题的计算复杂度，并有助于克服与有限数据和大型神经网络架构相关的挑战——这是算子学习中的常见障碍。 我们展示了该方法在各种边界条件下的泛化能力，并提供了其在数学物理领域潜在应用的见解。 我们的结果突显了 LE-NO 在捕捉复杂非线性行为方面的潜力，为发现和预测 PDEs 中的基础动力学提供了一个强大的工具。 显示英文摘要

摘要： Learning nonlinear partial differential equations (PDEs) is emerging in many scientific and engineering disciplines, driving advancements in areas such as fluid dynamics, materials science, and biological systems. In this work, we introduce the Laplacian Eigenfunction-Based Neural Operator (LE-NO), a framework for efficiently learning nonlinear terms in PDEs, with a particular focus on nonlinear parabolic equations. By leveraging data-driven approaches to model the right-hand-side nonlinear operator, the LE-NO framework uses Laplacian eigenfunctions as basis functions, enabling efficient approximation of the nonlinear term. This approach reduces the computational complexity of the problem by enabling direct computation of the inverse Laplacian matrix and helps overcome challenges related to limited data and large neural network architectures--common hurdles in operator learning. We demonstrate the capability of our approach to generalize across various boundary conditions and provide insights into its potential applications in mathematical physics. Our results highlight the promise of LE-NO in capturing complex nonlinear behaviors, offering a robust tool for the discovery and prediction of underlying dynamics in PDEs.