Identifiability of a dissipative knowledge-dynamics model: exact recovery under designed excitation, degeneration on observational data

Abstract

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 $\rho = 0.83$, 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.