離散オイラー・ラグランジュ方程式による構造保存ガウス過程
Structure-Preserving Gaussian Processes Via Discrete Euler-Lagrange Equations
離散強制オイラー・ラグランジュ方程式を用いたラグランジュガウス過程（LGPs）を提案し、位置データのみから物理的に一貫した力学系の学習と不確実性定量化を実現した。
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著者: Jan-Hendrik Ewering, Kathrin Flaßkamp, Niklas Wahlström, Thomas B. Schön, Thomas Seel
分類: cs.LG, cs.RO
原文アブストラクト
In this paper, we propose Lagrangian Gaussian Processes (LGPs) for probabilistic and data-efficient learning of dynamics via discrete forced Euler-Lagrange equations. Importantly, the geometric structure of the Lagrange-d'Alembert principle, which governs the motion of dynamical systems, is preserved by construction in the absence of external forces. This allows learning physically consistent models that overcome erroneous drift in the system's energy, thereby providing stable long-term predictions. At the core of our approach lie linear operators for Gaussian process conditioning, constructed from discrete forced Euler-Lagrange equations and variational discretization schemes. Thereby and unlike prior work, the method enables learning dynamics from discrete position snapshots, i.e., without access to a system's velocities or momenta. This is particularly relevant for a large class of practical scenarios where only position measurements are available, for instance, in motion capture or visual servoing applications. We demonstrate the data-efficiency and generalization capabilities of the LGPs in various synthetic and real-world case studies, including a real-world soft robot with hysteresis. The experimental results underscore that the LGPs learn physically consistent dynamics with uncertainty quantification solely from sparse positional data and enable stable long-term predictions.
