Identifiability of a dissipative knowledge-dynamics model: exact recovery under designed excitation, degeneration on observational data
Researchers prove a dissipative learning model is identifiable only under designed excitation, not observational data.
The paper models human learning as a nonlinear dissipative ODE system with a concept-transfer matrix, per-concept forgetting rates, and a saturating practice-response gain. It proves structural identifiability under designed excitation, including closed-form recovery for two concepts, and introduces a batched L-stable solver roughly two orders of magnitude faster on cohorts of 100,000 learners. Under the theorem's excitation conditions, synthetic parameters recover to machine precision and prerequisite structure at F1 = 1.0. An apparent Spearman ρ = 0.83 link between forgetting rates and topic difficulty on observational data fails controls that destroy temporal order, matching the predicted unidentifiable regime.
- Proves structural identifiability under designed excitation, with closed-form two-concept recovery.
- Batched L-stable solver is about 100x faster with bit-exact predictions.
- Synthetic recovery reaches machine precision and prerequisite F1 of 1.0.
- Observational Spearman 0.83 signal fails temporal-order and timestamp controls.
Full article251 words · extracted from arxiv.org · click to collapse
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.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2610.09889