用数据学习一维系统的格林函数，实现参数未见时的精准插值。
chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Data
- 基于有理神经网络与切比雪夫基构建连续格林函数模型。
- 在未知参数下通过奇异函数插值得到新解，误差可控。
- 适合物理建模中缺乏方程但有数据的场景，如材料力学。
本文提出一种无网格、数据驱动的方法 chebgreen，用于建模具有控制参数的一维系统，其偏微分方程未知。该方法从数据中学习隐含边值问题的实验格林函数，以有理神经网络形式表示，并构建在切比雪夫基下的双变量表示。通过在由拟矩阵构成的流形上插值左右奇异函数，实现对未见过控制参数值下的格林函数的重建；对应的奇异值则使用拉格朗日多项式进行插值。
原文摘要 · Abstract (English)
In this work, we present a mesh-independent, data-driven library, chebgreen, to mathematically model one-dimensional systems, possessing an associated control parameter, and whose governing partial differential equation is unknown. The proposed method learns an Empirical Green's Function for the associated, but hidden, boundary value problem, in the form of a Rational Neural Network from which we subsequently construct a bivariate representation in a Chebyshev basis. We uncover the Green's function, at an unseen control parameter value, by interpolating the left and right singular functions within a suitable library, expressed as points on a manifold of Quasimatrices, while the associated singular values are interpolated with Lagrange polynomials.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。