提出无需依赖网格的多保真度微分方程算子学习方法
Discretization-independent multifidelity operator learning for partial differential equations
- 用神经网络学习输入输出函数的基表示，实现离散化无关的算子建模
- 在局部与非局部PDE上验证，多保真训练显著提升精度和效率
- 首次系统揭示离散化无关性对多保真学习的关键作用，适合工程模拟场景
我们提出一种新的通用编码-近似-重构算子学习模型，利用神经网络学习输入与输出函数分布的基表示。引入了‘数值算子学习’与‘离散化无关’概念，厘清了算子学习理论与实际实现的关系。所提模型具有离散化无关性，特别适用于多保真度学习。我们在强假设下建立理论逼近保证，证明了统一普遍逼近性；在弱条件下获得统计逼近性。据我们所知，这是首个系统研究离散化无关性如何促进鲁棒高效多保真度算子学习的工作。通过大量数值实验验证，涵盖局部与非局部偏微分方程（PDE），包括时不变与时变问题。结果表明，多保真训练显著提升准确率与计算效率，并进一步增强经验上的离散化无关性。
原文摘要 · Abstract (English)
We develop a new and general encode-approximate-reconstruct operator learning model that leverages learned neural representations of bases for input and output function distributions. We introduce the concepts of \textit{numerical operator learning} and \textit{discretization independence}, which clarify the relationship between theoretical formulations and practical realizations of operator learning models. Our model is discretization-independent, making it particularly effective for multifidelity learning. We establish theoretical approximation guarantees, demonstrating uniform universal approximation under strong assumptions on the input functions and statistical approximation under weaker conditions. To our knowledge, this is the first comprehensive study that investigates how discretization independence enables robust and efficient multifidelity operator learning. We validate our method through extensive numerical experiments involving both local and nonlocal PDEs, including time-independent and time-dependent problems. The results show that multifidelity training significantly improves accuracy and computational efficiency. Moreover, multifidelity training further enhances empirical discretization independence.
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。