用深度瑞兹方法求解测地线，高效计算最优路径。
Finding geodesics with the Deep Ritz method
- 将测地线问题转化为变分优化，用神经网络逼近最优路径。
- 在路径规划、光学等四类问题中验证了方法有效性。
- 适合对物理建模与机器学习交叉研究感兴趣的读者。
测地线问题旨在计算从预设起点到终点的轨迹，以最小化用户定义的距离、代价或能量。这类问题广泛存在于物理和工程领域，例如复杂环境中的最优路径规划、折射介质中的光传播建模，以及控制理论和广义相对论中的时空轨迹研究。尽管其应用普遍，科学机器学习（SciML）社区对此类问题的研究仍较少。本文认为，由于测地线问题具有简单的几何结构、变分性质及自然非线性，特别适合采用深度瑞兹方法。我们通过四个数值案例——路径规划、光学、固体力学和生成建模——验证了该方法的潜力。本研究目标并非全面覆盖测地线问题，而是揭示深度瑞兹方法的一个有前景的应用方向，为未来科学机器学习研究提供启示。
原文摘要 · Abstract (English)
Geodesic problems involve computing trajectories between prescribed initial and final states to minimize a user-defined measure of distance, cost, or energy. They arise throughout physics and engineering -- for instance, in determining optimal paths through complex environments, modeling light propagation in refractive media, and the study of spacetime trajectories in control theory and general relativity. Despite their ubiquity, the scientific machine learning (SciML) community has given relatively little attention to investigating its methods in the context of these problems. In this work, we argue that given their simple geometry, variational structure, and natural nonlinearity, geodesic problems are particularly well-suited for the Deep Ritz method. We substantiate this claim with four numerical examples drawn from path planning, optics, solid mechanics, and generative modeling. Our goal is not to provide an exhaustive study of geodesic problems, but rather to identify a promising application of the Deep Ritz method and a fruitful direction for future SciML research.
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。