# A zero-one law for one-shot system identification

arXiv: https://arxiv.org/abs/2607.15832 · HTML: https://mlsanity.com/paper/2607.15832/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 27 references that are in the map.
- First posted on arXiv 17 Jul 2026.

## Builds on (most cited first, 5 of 5)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Elliptic PDE learning is provably data-efficient](https://mlsanity.com/paper/2302.12888/) (2023)
- [Fourier Weak SINDy: Spectral Test Function Selection for Robust Model Identification](https://mlsanity.com/paper/2604.20141/) (2026)

## Abstract (arXiv)

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.