提出可学习收缩积分算子的新模型，解决高维积分方程求解难题。
Learning Contractive Integral Operators with Fredholm Integral Neural Operators
- 基于弗雷德霍姆积分方程构建神经算子，保证算子收缩性
- 在任意维度上实现线性和非线性积分方程的高精度逼近
- 适用于科学计算中的偏微分方程求解，结果可解释
我们推广了弗雷德霍姆神经网络框架，用于学习任意维度下第二类弗雷德霍姆积分方程（FIEs）中出现的非扩张积分算子。提出弗雷德霍姆积分神经算子（FREDINOs），证明其可通用逼近线性与非线性积分算子及其解算子，并确保学习到的算子具有收缩性，严格满足不动点迭代收敛所需的数学性质。此外，展示了FREDINOs通过边界积分方程（BIE）形式学习非线性椭圆型PDE解算子的能力。通过多个基准问题进行数值验证：任意维度的线性与非线性FIEs，以及二维非线性椭圆型PDE。基于定制化的数学与数值分析理论，FREDINOs提供高精度近似与可解释的计算方案，非常适合科学机器学习与数值分析应用。
原文摘要 · Abstract (English)
We generalize the framework of Fredholm Neural Networks, to learn non-expansive integral operators arising in Fredholm Integral Equations (FIEs) of the second kind in arbitrary dimensions. We first present the proposed Fredholm Integral Neural Operators (FREDINOs), for FIEs and prove that they are universal approximators of linear and non-linear integral operators and corresponding solution operators. We furthermore prove that the learned operators are guaranteed to be contractive, thereby strictly satisfying the mathematical property required for the convergence of the fixed point scheme. Finally, we also demonstrate how FREDINOs can be used to learn the solution operator of non-linear elliptic PDEs, via a Boundary Integral Equation (BIE) formulation. We assess the proposed methodology numerically, via several benchmark problems: linear and non-linear FIEs in arbitrary dimensions, as well as a non-linear elliptic PDE in 2D. Built on tailored mathematical/numerical analysis theory, FREDINOs offer high-accuracy approximations and interpretable schemes, making them well suited for scientific machine learning/numerical analysis computations.
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。