耗散知识动力学模型的可辨识性：设计激励下的精确恢复与观测数据上的退化

中文摘要 AI 辅助

人类学习是一个耗散动力学过程：熟练度通过练习而积累，通过遗忘而衰减，并在相互依赖的概念之间传播。我们将其建模为一个非线性耗散常微分方程组，其参数具有机制层面的意义（一个编码先决条件耦合的概念转移矩阵、各概念的遗忘率以及一个饱和的练习响应增益），并研究这些参数何时能真正从数据中恢复。我们证明了在显式激励条件下相关反问题的结构可辨识性定理，给出了两概念情形下构造性的闭式恢复方法，以及单调性、鲁棒性和L稳定性结果。我们为耗散子系统推导了一个半隐式L稳定格式，以及一个在数值上等价于逐轨迹公式（位精确预测，梯度精确到$10^{-10}$）但速度快两个数量级的批处理求解器，使得在$10^5$名学习者的队列上进行估计成为可行。实证研究是双面的。在定理的激励条件下，合成恢复是精确的：参数达到机器精度，先决条件结构的$F_1 = 1.0$。在大型观测基准上则不然。一个看似很强的恢复结果——遗忘率与主题难度相关，Spearman $\rho = 0.83$——被四个独立对照所驳斥：它经受住了破坏数据时间顺序的考验，与经典贝叶斯基线相当，并且不受移除真实时间戳的影响。我们将其追溯到模型的平稳结构，并表明这正是定理在缺乏设计激励时所预测的退化。该结果划定了可辨识与不可辨识机制之间的清晰边界，并为可解释性声明提供了一个验证协议。

英文摘要

Human learning is a dissipative dynamical process: mastery accumulates through practice, decays through forgetting, and propagates across interdependent concepts. We model it as a nonlinear dissipative system of ordinary differential equations whose parameters are mechanistically meaningful (a concept-transfer matrix encoding prerequisite coupling, per-concept forgetting rates, and a saturating practice-response gain), and we study when those parameters can actually be recovered from data. We prove a structural identifiability theorem for the associated inverse problem under explicit excitation conditions, with constructive closed-form recovery for the two-concept case, together with monotonicity, robustness and L-stability results. We derive a semi-implicit L-stable scheme for the dissipative subsystem and a batched solver numerically equivalent to the per-trajectory formulation (bit-exact predictions, gradients to $10^{-10}$) yet two orders of magnitude faster, making estimation feasible on cohorts of $10^5$ learners. The empirical study is two-sided. Under the theorem's excitation conditions, synthetic recovery is exact: parameters to machine precision, prerequisite structure at $F_1 = 1.0$. On large observational benchmarks it is not. An apparently strong recovery, with forgetting rates correlating with topic difficulty at Spearman $ρ= 0.83$, is refuted by four independent controls: it survives destroying the temporal order of the data, is matched by a classical Bayesian baseline, and is unaffected by removing real timestamps. We trace this to the stationary structure of the model and show that it is the degeneration the theorem predicts in the absence of designed excitation. The result delineates a sharp boundary between identifiable and unidentifiable regimes and yields a validation protocol for interpretability claims.

发表机构

• Innopolis University（英诺波利斯大学）
• HSE University（高等经济大学）

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