クープマン演算子と,シンボリック・ダイナミクスによる最大エントロピーの測定
Connor Kennedy1, John Kaushagen2, Hong-Kun Zhang3
1Department of Mathematics, Brandeis University, Waltham, Massachusetts 02453, USA.
Chaos (Woodbury, N.Y.)
|February 17, 2026
まとめ
シンボリック拡張ダイナミックモード分解 (Symbolic Extended Dynamic Mode Decomposition,EDMD) は,現在,不変数測定を必要とせずに,クープマン演算子を推定しています. この方法では,特定の動的システムの最大エントロピーの測定値も近似されます.
科学分野:
- ダイナミック・システム理論
- エルゴディック理論 エルゴディック理論
- データ駆動科学とは,データ駆動科学です.
背景:
- シンボリック拡張動態分解 (Symbolic Extended Dynamic Mode Decomposition,EDMD) は,以前,不変の測定値と既知のパーティションを使用してコップマン演算子を推定していました.
- 制限には,システム特性の事前の知識の要求が含まれていた.
研究 の 目的:
- 恒常的な措置の必要性を排除することによって,象徴的なEDMDを前進させる.
- 方法が最大エントロピー (MME) の測定値に近似していることを示します.
主な方法:
- インヴァリアントの測定に関する知識を必要としない新しい象徴的なEDMDフレームワークを開発しました.
- この方法により,マルコフまたはソフィックのシンボリックシフトのMMEを近似する一連の測定値が生成されることが証明された.
- この方法をリヴァラニ・サウッソル・ヴァイエンティの地図,マルコフ以外の断片的線形地図,アーノルド・キャット・マップに適用した.
主要な成果:
- テストされたダイナミックシステムのスペクトルデータとMMEの推定を成功させた.
- 異なるシステムタイプで象徴的なEDMDメソッドの堅実性を実証しました.
- シンボリックシリンダーセットの推定のための新しいデータ駆動技術,特にキャットマップを導入しました.
結論:
- 強化されたシンボリックEDMDメソッドは,インヴァリアントの測定に関する事前の知識なしに,ダイナミックシステムを分析するための強力なツールです.
- この方法は,MMEを近似するためにデータベースのアプローチを提供し,エルゴディック理論における理論的および計算的研究を進めています.
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