用几何方法学习低维物理模型，提升复杂系统预测精度与数据效率。
A Riemannian Framework for Learning Reduced-order Lagrangian Dynamics
- 基于黎曼几何构建可保持物理规律的低维隐空间
- 在少数据下实现刚体与柔体系统的长期精准预测
- 适合需要高效、可解释动力学建模的研究者
通过引入物理一致性作为归纳偏置，深度神经网络在学习非线性动态模型时表现出更强的泛化能力与数据效率。然而，这类模型的复杂度通常随系统维度增加而上升，需更大数据集、更复杂的网络结构和更高的计算成本。本文提出一种新型几何网络架构，用于学习能准确描述高维系统行为的物理一致的低维拉格朗日参数。该方法结合模型降阶最新进展，从黎曼视角联合学习非线性保结构的隐空间及其对应的低维动力学。所提方法可在减少数据需求的前提下，实现刚体与柔体系统高维动态的高精度长期预测，并推导出可解释且物理合理的低维拉格朗日模型。
原文摘要 · Abstract (English)
By incorporating physical consistency as inductive bias, deep neural networks display increased generalization capabilities and data efficiency in learning nonlinear dynamic models. However, the complexity of these models generally increases with the system dimensionality, requiring larger datasets, more complex deep networks, and significant computational effort. We propose a novel geometric network architecture to learn physically-consistent reduced-order dynamic parameters that accurately describe the original high-dimensional system behavior. This is achieved by building on recent advances in model-order reduction and by adopting a Riemannian perspective to jointly learn a non-linear structure-preserving latent space and the associated low-dimensional dynamics. Our approach enables accurate long-term predictions of the high-dimensional dynamics of rigid and deformable systems with increased data efficiency by inferring interpretable and physically-plausible reduced Lagrangian models.
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