Model Error Resonance: The Geometric Nature of Error Dynamics

TL;DR

This paper develops a geometric framework for understanding model error dynamics, revealing how curvature mismatch influences error evolution and resonance phenomena in both continuous and discrete systems.

Contribution

It introduces a novel geometric theory of model error using affine connections and curvature, linking error dynamics to intrinsic manifold properties and extending to discrete systems.

Findings

Error dynamics are governed by curvature mismatch as a Jacobi-type equation.

Model Error Resonance occurs under positive curvature in flat model connections.

Curvature can be inferred from error evolution in a sphere-plane example.

Abstract

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.