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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.11734 |
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| _version_ | 1866909958762659840 |
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| author | Dai, Yuntao |
| author_facet | Dai, Yuntao |
| contents | This paper introduces a geometric theory of model error, treating true and model dynamics as geodesic flows generated by distinct affine connections on a smooth manifold. When these connections differ, the resulting trajectory discrepancy--termed the Latent Error Dynamic Response (LEDR)--acquires an intrinsic dynamical structure governed by curvature. We show that the LEDR satisfies a Jacobi-type equation, where curvature mismatch acts as an explicit forcing term. In the important case of a flat model connection, the LEDR reduces to a classical Jacobi field on the true manifold, causing Model Error Resonance (MER) to emerge under positive sectional curvature. The theory is extended to a discrete-time analogue, establishing that this geometric structure and its resonant behavior persist in sampled systems. A closed-form analysis of a sphere--plane example demonstrates that curvature can be inferred directly from the LEDR evolution. This framework provides a unified geometric interpretation of structured error dynamics and offers foundational tools for curvature-informed model validation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11734 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Model Error Resonance: The Geometric Nature of Error Dynamics Dai, Yuntao Systems and Control Dynamical Systems This paper introduces a geometric theory of model error, treating true and model dynamics as geodesic flows generated by distinct affine connections on a smooth manifold. When these connections differ, the resulting trajectory discrepancy--termed the Latent Error Dynamic Response (LEDR)--acquires an intrinsic dynamical structure governed by curvature. We show that the LEDR satisfies a Jacobi-type equation, where curvature mismatch acts as an explicit forcing term. In the important case of a flat model connection, the LEDR reduces to a classical Jacobi field on the true manifold, causing Model Error Resonance (MER) to emerge under positive sectional curvature. The theory is extended to a discrete-time analogue, establishing that this geometric structure and its resonant behavior persist in sampled systems. A closed-form analysis of a sphere--plane example demonstrates that curvature can be inferred directly from the LEDR evolution. This framework provides a unified geometric interpretation of structured error dynamics and offers foundational tools for curvature-informed model validation. |
| title | Model Error Resonance: The Geometric Nature of Error Dynamics |
| topic | Systems and Control Dynamical Systems |
| url | https://arxiv.org/abs/2512.11734 |