单次实验能否识别系统模型？答案要么全不行，要么几乎总行。
A zero-one law for one-shot system identification
- 基于字典项线性独立性判断能否单次恢复模型
- 证明了非退化高斯输入下几乎必然可恢复
- 适合需要快速建模且数据稀缺的场景
能否仅通过一次实验识别模型？我们研究线性参数化于预设字典项（如偏微分算子、动力系统）的解析系统。对于单一输入-响应对，恢复成立当且仅当所评估的字典项线性无关。我们证明了一个精确的零一法则：要么无输入能唯一确定系数，要么几乎所有从非退化高斯测度采样的随机输入均可。该二分法将单次系统辨识简化为对退化输入的分析，并提供任何恢复模型的后验验证证书。数值例子从单条轨迹数据中成功恢复动力系统、非线性偏微分方程和结构矩阵族，同时检测出何时需额外探测。
原文摘要 · Abstract (English)
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.
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