analiza aiqbench
This research establishes a fundamental theoretical boundary in system identification by proving that recovering linearly parameterized models from a single experiment is either impossible or succeeds with probability one, depending solely on the independence of dictionary terms. By reducing complex recovery problems to questions about degenerate inputs and providing an *a posteriori* certificate for recovered models, this work offers rigorous guarantees essential for training robust AI agents using minimal data trajectories. The findings directly advance machine learning theory regarding sample efficiency and model identifiability in dynamical systems.
A zero-one law for one-shot system identification
arXiv:2607.15832v1 Announce Type: cross 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.
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.
Numerical Analysis (math.NA); Machine Learning (cs.LG); Dynamical Systems (math.DS)
arXiv:2607.15832 [math.NA]
(or arXiv:2607.15832v1 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2607.15832 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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