物理信息学习中神经表示与样条表示的比较

中文摘要 AI 辅助

物理信息机器学习近来作为一种求解偏微分方程的新范式出现，其功能包括参数化建模和逆向设计，使用神经网络来表示解场。然而，已有若干关于此类网络训练困难的报道，导致收敛缓慢、解精度低以及对超参数选择敏感。为了更好地理解这些困难背后的机制，本工作提出在同一物理信息学习框架内对神经网络和B样条表示进行严格比较。通过数值研究四个具有闭式解的问题，从渐近收敛率、采样需求、损失函数最小化以及所需算术精度等方面，对两种表示的精度和效率进行严格分析。

英文摘要

Physics-informed machine learning as recently emerged as a new paradigm for the resolution of partial differential equations, including functionalities for parametric modeling and inverse design, using neural networks to represent the solution fields. However, several difficulties have been reported concerning the training of such networks, resulting in slow convergence, low accurate solutions and sensitivity to the choice of hyper-parameters. In order to better understand the mechanisms underlying these difficulties, the present work proposes a rigourous comparison of neural networks and B-Splines representations within the same physics-informed learning framework. Four problems with closed-form solutions are numerically studied to establish a rigorous analysis of the accuracy and efficiency of the two representations, in terms of asymptotic convergence rates, sampling requirements, loss function minimization and arithmetic precision needed.

发表机构

• Université Côte d’Azur（蔚蓝海岸大学）
• Inria（法国国家信息与自动化研究所）
• CNRS（法国国家科学研究中心）
• LJAD（约瑟夫·傅里叶分析、几何及其应用实验室）

机构由 AI 辅助整理，请以论文原文为准。