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 105 learners. The empirical study is two-sided. Under the theorem's excitation conditions, synthetic recovery is exact: parameters to machine precision, prerequisite structure at F1=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.
Figures & tables
h , days
hΛmax
explicit RK4
scheme ( 15 )
error
box
error
box
0.25
0.50
4.1⋅10−11
ok
1.3⋅10−3
ok
1.00
2.00
1.1⋅10−8
ok
4.9⋅10−3
ok
2.00
4.00
1.7⋅1010
escape
9.8⋅10−3
ok
5.00
10.0
3.3⋅1014
escape
2.3⋅10−2
ok
30.0
60.0
2.8⋅105
escape
1.1⋅10−1
ok
Table 1: Absolute error after a gap of T=30 days. “escape” indicates that the solution left the invariant box [0,1] .
Dataset
Cognitive-PINN
DKT
BKT
ASSISTments-2009
0.682
0.758
0.717
ASSISTments-2015
0.695
0.730
0.691
ASSISTments-2017
0.607
0.697
0.627
Table 2: Matched AUC on contamination-controlled benchmarks. All pairwise differences are significant (DeLong, p≈0 ).
Figure 1: Recovered forgetting rates versus independently measured topic difficulty on Junyi ( K=39 ), across five seeds. The correlation is strong and stable — and, as Section 7.4 shows, not evidence of recovering forgetting.
Control
Quantity
Result
C1 shuffle
ρ(Λ,difficulty) after permuting order
0.829 (vs 0.845 )
C2 baseline
ρ(BKT slip,difficulty)
0.795
ρ(BKT guess,difficulty)
−0.699
C3 real time
ρ(Λ,empirical forgetting slope)
0.49±0.01
C4 time ablation
same, real time removed
0.504
Table 3: Controls on the apparent parameter recovery.
Figure 2: Structural recovery across five seeds against the null distribution of random graphs of identical density. The observed edge-AUC values lie close to the upper tail of the null, and the spread across seeds exceeds the margin over chance.
Learning governing dynamics from data is a common goal across the sciences, yet it is only well-posed when the underlying mechanisms are identifiable. In practice, many data-driven methods implicitly assume identifiability; when this assumption fails, estimated models can yield spurious predictions and invalid mechanistic conclusions. Classical identifiability guarantees for controlled linear time-invariant (LTI) systems provide sufficient conditions -- controllability and persistent excitation -- but leave open whether identifiability holds when these conditions fail, and which parts of the system remain identifiable without full identifiability. We show that the experimental setup, i.e., the realized initial state and control input, dictates a fundamental limit on the information recoverable from the observed trajectory. We develop a geometric characterization of this limit and derive a closed-form description of all systems consistent with the experimental setup. Crucially, we prove that even when the full system is not identifiable, the restricted dynamics on the subspace reachable by the experiment remain uniquely determined.
Aybüke Ulusarslan, Niki Kilbertus, Nora Schneider
1Technical University of Munich · 2Helmholtz Munich · 3Munich Center for Machine Learning (MCML)
A wide range of methods have been proposed, including physics-informed neural networks, which are powerful but do not guarantee identifiability of the dynamics, symbolic regression, which requires a set of precomputed operations, and causal discovery, which is more principled but usually relies on strong assumptions that physical systems may violate. In this work, we develop a theory-grounded method and prove that under a set of permissive assumptions, the structural drivers and drift of stochastic delayed differential equations are identifiable. Our method outperforms others on a benchmark for driver identifiability, and on a second benchmark to evaluate physical consistency of the learned dynamics.
Julien Boussard, Antoine Debouchage, Théo Saulus
School of Computer Science, McGill University, Canada · Mila - Quebec AI Institute, Canada · LaMMe, Université Évry Paris-Saclay, France +1
We consider the problem of learning the parameters of a N-dimensional stochastic linear dynamics under both full and partial observations from a single trajectory of time T. We introduce and analyze a new estimator that achieves a small maximum element-wise error on the recovery of symmetric dynamic matrices using only T=O(logN) observations, irrespective of whether the matrix is sparse or dense. This estimator is based on the method of moments and does not rely on problem-specific regularization. This is especially important for applications such as structure discovery.
Minh Vu, Andrey Y. Lokhov, Marc Vuffray
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA