从离散观测中求解常微分方程反问题的一般局部可识别性
Generic local identifiability for ODE inverse problems from discrete observations
AI总结：
研究常微分方程模型参数局部可识别性，通过对解采样得参数到观测映射，证明\(C^2\)观测映射定量单射估计，处理解析模型得一般局部可识别性，针对齐次线性系统分离\(e^{hA}\)与\(A\)恢复，确定离散传播子及剩余模糊性。
摘要：
我们研究常微分方程模型中参数的局部可识别性，通过在规定时间对解进行采样得到参数到观测的映射。首先证明一般\(C^2\)观测映射的定量单射估计，给出参数雅可比最小奇异值下界及二阶导数上界，确定局部解唯一稳定的邻域。接着处理解析常微分方程模型，在单一非退化条件下得到一般局部可识别性。最后针对齐次线性系统\(\dot X = AX\)，将\(e^{hA}\)与\(A\)的恢复分离，循环初始状态可识别离散传播子，剩余模糊性恰为实矩阵对数的模糊性。
英文摘要：
We study local identifiability of parameters in ordinary differential equation models from finitely many observations. The central object is the parameter-to-observation map obtained by sampling the solution at prescribed times. We first prove a quantitative injectivity estimate for general $C^2$ observation maps: a lower bound on the smallest singular value of the parameter Jacobian, together with an upper bound on the second derivative, gives an explicit neighborhood on which the inverse problem has a unique and stable local solution. We then treat analytic ODE models. For analytic vector fields the observation map is analytic in observation times, initial states, and parameters; consequently, the loss of full parameter rank is contained in the zero set of a real analytic function. Under a single non-degeneracy condition this gives generic local identifiability, including for randomly chosen observation times with a density. Finally, for homogeneous linear systems $\dot X=AX$, we separate the recovery of $e^{hA}$ from the recovery of $A$: cyclic initial states identify the discrete propagator from one trajectory, while the remaining ambiguity is precisely the ambiguity of the real matrix logarithm.
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