A zero-one law for one-shot system identification

2026-07-17

AI Summary

This work investigates whether an analytic system model, linearly parameterized over a prescribed dictionary (e.g., partial differential operators or dynamical system terms), can be uniquely identified from a single input–response trajectory. By analyzing the linear independence of dictionary elements under a single observation, the authors establish and prove a sharp zero–one law: either no input enables unique identification, or almost every input drawn from a non-degenerate Gaussian measure permits exact recovery. This result reframes one-shot system identification as a problem of detecting degenerate inputs and provides a posteriori verification. Combining tools from linear algebra, measure theory, and system identification, the approach successfully reconstructs dynamical systems, nonlinear partial differential equations, and structured matrix families from a single trajectory, while accurately determining whether additional probing signals are necessary.

Abstract

Can a model be identified from one experiment? We study analytic systems that are linearly parameterized by a combination of prescribed dictionary terms, such as partial differential operators and dynamical systems. For a single input-response pair, recovery is possible exactly when the evaluated dictionary terms are linearly independent. We prove a sharp zero-one law: either no input uniquely determines the coefficients, or almost every random input sampled from a nondegenerate Gaussian measure does. This dichotomy reduces one-shot system identification to a question about degenerate inputs and provides an a posteriori certificate for any recovered model. Numerical examples recover dynamical systems, nonlinear partial differential equations, and structured matrix families from single trajectory data, while also detecting when an extra probe is necessary.

Problem

• one-shot system identification
• zero-one law
• linear independence
• analytic systems
• model recovery

Innovation

• zero-one law
• one-shot system identification
• linear parameterization
• dictionary-based recovery
• Gaussian measure