弱形式方法在方程学习中表现出强噪声鲁棒性与高效计算能力。
The Weak Form Is Stronger Than You Think
- 利用弱形式重构方程，提升对噪声数据的适应性。
- 在参数估计与粗粒化中实现高精度与低计算开销。
- 适合关注数值稳定性与计算效率的研究者参考。
弱形式是现代计算与应用数学中普遍、深入研究并广泛应用的数学工具。本文综述了弱形式在多个领域中的历史发展与最新进展，重点介绍其在方程学习、参数估计和粗粒化中的若干新突破，这些方法展现出惊人的噪声鲁棒性、高精度与计算效率。本文为2024年10月发表于SIAM News同名文章的补充，提供更详尽的数学推导与更完整的参考文献列表。文中结果可复现代码已发布于课题组GitHub：https://github.com/MathBioCU。
原文摘要 · Abstract (English)
The weak form is a ubiquitous, well-studied, and widely-utilized mathematical tool in modern computational and applied mathematics. In this work we provide a survey of both the history and recent developments for several fields in which the weak form can play a critical role. In particular, we highlight several recent advances in weak form versions of equation learning, parameter estimation, and coarse graining, which offer surprising noise robustness, accuracy, and computational efficiency. We note that this manuscript is a companion piece to our October 2024 SIAM News article of the same name. Here we provide more detailed explanations of mathematical developments as well as a more complete list of references. Lastly, we note that the software with which to reproduce the results in this manuscript is also available on our group's GitHub website https://github.com/MathBioCU .
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