Identifiability Guarantees for Drivers and Dynamics of Delayed Physical Systems

Julien Boussard; Antoine Debouchage; Théo Saulus

stat.MLSep 29, 2026

Identifiability Guarantees for Drivers and Dynamics of Delayed Physical Systems

Authors: Julien Boussard, Antoine Debouchage, Théo Saulus

Organizations: School of Computer Science, McGill University, Canada · Mila - Quebec AI Institute, Canada · LaMMe, Université Évry Paris-Saclay, France · DIRO, Université de Montréal, Canada

Abstract

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.

Figures & tables

	Experiment	Simple			Coupled					Climate		Avg. rank
	Experiment	Default	Confounder	Noise	Default	Noise	Confounder	Time-lag	Standardize	MAOOAM	ENSO	Avg. rank
SHD	PCMCI+	41.04	23.02	47.64	224.80	183.90	324.63	327.72	228.32	80.00	529.36	7.30
	F-PCMCI	35.30	21.07	45.09	192.90	149.60	195.74	350.61	201.79	130.00	530.27	5.80
	NGC	28.91	19.96	28.84	840.95	842.55	670.53	793.67	840.26	31.00	337.09	7.90
	CUTS+	48.11	24.04	61.32	152.00	150.50	272.68	247.22	310.63	130.00	608.73	7.50
	RCD	61.85	26.74	61.38	157.05	155.65	136.53	201.11	159.84	130.00	665.36	6.80

Table 1:

L_{0}

-NDDE outperforms competing methods on the CausalDynamics benchmark. SHD (lower is better) and AUROC / AUPRC (higher is better) are reported for

L_{0}

-NDDE (Ours), AGL, C-NODE and the benchmark causal discovery methods. The best values are bold, the second best values are underlined. ∗ indicates statistical significance between AGL or C-NODE and

L_{0}

-NDDE, calculated with a Welch’s t-test (p-value 0.05). The benchmark only provides the mean values for the causal discovery methods, and we thus do not run statistical significance tests for these methods. For AUROC and AUPRC, statistical significance results always match, so we report only one ∗ for each pair. The last column reports the average rank for each method and metric. For readability, methods that never score best are not shown here, but in the Appendix ( Table 4 )

Method	NRMSE ( $\downarrow$ )		VLPT ( $\uparrow$ )		FLEE ( $\downarrow$ )		$D_{KY}$ ( $\downarrow$ )		LSD ( $\downarrow$ )		$D_{stsp}$ ( $\downarrow$ )		SHD ( $\downarrow$ )
Method	Med.	Rank	Med.	Rank	Med.	Rank	Med.	Rank	Med.	Rank	Med.	Rank	Med.	Rank
AGL	1.15	2.69	2.85	3.14	0.25	3.47	0.88 ∗	3.82	14.71 ∗	2.82	3e 6 ∗	4.37	0.44 ∗	4.46
PathReg	1.19	2.49	4.33	3.24	0.25	3.78	0.89 ∗	3.9	16.99	3.02	3e 6 ∗	4.62	0.22	2.70
C-NODE	1.53	4.05	4.61	3.39	0.20	3.00	0.72 ∗	3.17	37.79 ∗	4.08	34.60 ∗	2.86	0.17	2.67
NODE	1.20	2.94	5.12	2.63	0.16	2.51	0.076	2.17	13.99	2.57	4.29	1.62	0.22	2.70
$L_{0}$ -NDDE	1.19	2.83	6.54	2.60	0.16	2.25	0.075	1.94	14.75	2.50	2.50	1.54	0.11	2.45

Table 2:

L_{0}

-NDDE outperforms other feature selection methods.

L_{0}

-NDDE ranks best in five of the metrics. The difference with NODE is non significant, as NODE also performs very well, but

L_{0}

-NDDE consistently ranks better than the other methods performing input feature selection, especially on attractor dimension reconstruction (

D_{KY}

) and convergence in distribution (

D_{stsp}

). ∗ indicates statistical significance with the

L_{0}

-NDDE metric, obtained using a permutation test (p-value 0.05). For each metric, best results are bolded and second best are underlined.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 1 — Figure 1: Dependency map of the theoretical results. Yellow boxes are assumptions, blue boxes are lemmas and propositions, and green boxes are theorems. Arrows denote when an assumption or a result is used in another proof. A use already implied by a chain of arrows is not drawn, unless it plays a separate role (labelled arrows). The proofs are grouped vertically into the three groups of proofs in Appendix A , contributing the finite-sample guarantee at the bottom.

Architecture, functions $f_{j}$
Parameter	Value
Number of layers	2
Number of units per layer	8
Optimization parameters
Gradient penalty coefficient	100
Warm-up iterations	100

Appendix

Table 3: Final hyperparameters used for

L_{0}

-NDDE, C-NODE, AGL and No grad. pen. The last 4 lines show the penalty coefficient for each method, as it is the only parameter tuned for each the method. Note that No grad. pen. does not use gradient penalty.

	Experiment	Simple			Coupled					Climate		Avg. rank
	Experiment	Default	Confounder	Noise	Default	Noise	Confounder	Time-lag	Standardize	MAOOAM	ENSO	Avg. rank
SHD	PCMCI+	41.04	23.02	47.64	224.80	183.90	324.63	327.72	228.32	80.00	529.36	7.30
	F-PCMCI	35.30	21.07	45.09	192.90	149.60	195.74	350.61	201.79	130.00	530.27	5.80
	VARLiNGAM	35.69	22.04	42.84	311.45	248.90	159.63	449.33	349.63	130.00	453.00	8.00
	DYNOTEARS	52.37	21.74	51.68	181.50	180.95	248.32	261.44	243.84	94.00	589.36	6.60
	NGC	28.91	19.96	28.84	840.95	842.55	670.53	793.67	840.26	31.00	337.09	7.90

Appendix

Table 4:

L_{0}

-NDDE outperforms competing methods most of the time on the CausalDynamics benchmark. This table is similar to Table 1 with additional methods being shown.

	Experiment	Simple			Coupled					Climate
	Experiment	Default	Confounder	Noise	Default	Noise	Confounder	Time-lag	Standardize	MAOOAM	ENSO
SHD	No Grad. Pen.	26.95	18.21	24.47	215.08	142.67	145.08	233.58	220.68	49.00	341.27
SHD	$L_{0}$ -NDDE (Ours)	26.42	16.25	23.67	205.35	144.20	143.33	222.34	212.11	40.00	325.54
AUROC /	No Grad. Pen.	.64 / .80	.58 / .65	.61 / .79	.63 / .42	.64 / .44	.67 / .47	.59 / .41	.65 / .44	.85 / .96	.52 / .68
AUPRC	$L_{0}$ -NDDE (Ours)	.64 / .80	.58 / .66	.63 / .80	.64 / .43	.65 / .44	.67 / .47	.60 / .43	.66 / .45	.86 / .96	.58 / .75

Appendix

Table 5:

L_{0}

-NDDE is slightly stronger with gradient penalty. Gradient penalty helps avoid overfitting spurious correlations and leads to slightly better performance on the CausalDynamics benchmark.

Optimization parameters	$L_{0}$ -NDDE	NODE	C-NODE	AGL	PathReg
Gradient penalty coefficient	0	100	0	1000	1000
niter_warmup	0	0	100	50	200
n_iter_after_convergence	200	0	500	1000	500
Sparsity penalty coefficients	0.05	0	0.05	50	1

Appendix

Table 6: Final hyperparameters used for

L_{0}

-NDDE, C-NODE, AGL and PathReg

Figure 2: Example predicted trajectories (red) vs. true trajectories (black) on 26 Dysts systems, for $L_{0}$ -NDDE (left), NODE (middle), and C-NODE (right).

Figure 3: Lower training error and lower SHD are both associated with higher performance. We perform an analysis to better understand why

L_{0}

-NDDE and NODE outperform other methods on the Dysts datasets. We compute, for all methods and all datasets, the training error (Train MSE) and SHD. We then plot the 6 metrics versus the SHD (upper half) and versus the training error (lower half). We see that

D_{KY}

and

D_{stsp}

increase, while VLPT decreases with SHD and training error. These relationships are statistically significant according to a Kendall-

\tau

test, a standard rank-correlation test for investigating associations between two sets of values, without assuming Gaussianity of the samples or needing large samples. The relationship is also significant between LSD and training error. This indicates that a lower training error is associated with better reconstruction, indicating why NODE performs well, as the absence of input feature regularization allows it to overfit to the training data; and why

L_{0}

-NDDE outperforms other methods, as it achieves lower SHD. Moreover, the

L_{0}

-penalty has the advantage that it does not perform weight decay, but explicitly turns features on or off, allowing the rest of the network to approximate the dynamics very well given the set of selected features.

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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.

Abstract

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