ジオメトリックニューラル普通微分方程式: マニフォールドからリー群へ

Yannik P Wotte¹, Federico Califano¹, Stefano Stramigioli¹

¹Robotics and Mechatronics, EEMCS, University of Twente (UT), Drienerlolaan 5, 7522 NB Enschede, The Netherlands.

Entropy (Basel, Switzerland)

|August 28, 2025

まとめ

ニューラル普通微分方程式 (ニューラル ODE) は微分多様体とリー群に拡張され,標準的なユークリッド空間を超えた複雑な動的システムの最適化を可能にします.

科学分野:

ダイナミック・システム
機械学習
微分幾何学

背景:

ニューラル普通微分方程式 (ニューラル ODE) は,ダイナミックシステムにおけるパラメータ最適化に使用されている.
ニューラルODEに関する既存の理論的結果は主にEuclidean space (Rn) に焦点を当てている.
多くの実世界のダイナミックシステムは,微分多様体やリー群のようなより複雑な空間で進化する.

研究の目的:

神経 ODE の理論的枠組みを微分多様体へと拡張する.
多数のニューラル ODE の既存の結果の統一派生を提供すること.
Lie グループで進化するシステムのためのニューラル ODE の拡張を導入する.

主な方法:

ニューラルODEに関する最近の理論的結果の収集と統一.
既存のニューラル ODE 方法を微分多様体に拡張するためのチュートリアルフレームワークの開発.
複数のニューラル ODE を拡張し,リー群の構造を利用する.

主要な成果:

微分可能な多様体におけるニューラル ODE の統一派生が提示される.
このフレームワークは様々なダイナミック・システムに適用可能であることが示されています.
マニポリドニューラル ODE の新しい拡張は,Lie グループのために開発され,それらの特定の構造を活用します.

結論:

神経ODEは,微分可能な多様体とリー群に効果的に一般化することができる.
この一般化は,より広範な複雑な動的システムにニューラル ODE の適用性を拡張します.
提示された方法は,幾何学的なディープラーニングと物理情報に基づいたニューラルネットワークの将来の研究のための経路を提供します.

キーワード:

リーグループ微分幾何学機械学習ニューラル普通微分方程式最適な制御

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