On Parameters of Nonlinear Scalar Dynamics from Video: Invariants, Calibration, and Identifiability

Wenjie Wang; Yuanyuan Wang; Zixiang Jiang; Shaoan Xie; Mingming Gong

cs.LGSep 30, 2026

On Parameters of Nonlinear Scalar Dynamics from Video: Invariants, Calibration, and Identifiability

Authors: Wenjie Wang, Yuanyuan Wang, Zixiang Jiang, Shaoan Xie, Mingming Gong

Organizations: University of Melbourne · Mohamed bin Zayed University of Artificial Intelligence · Carnegie Mellon University

Abstract

Physical parameter estimation from video aims to recover the parameters of a known family of governing dynamical equations from pixel observations. Existing identifiability theory for this setting has focused on linear time-invariant (LTI) second-order systems, leaving open what can be identified for nonlinear scalar dynamics. We develop an identifiability theory for nonlinear scalar second-order ODEs, organized by how their velocity dependence interacts with changes of the learned state coordinate. Under a shared non-collapsed state map and explicit same-state velocity-coverage conditions, we show that parameter identifiability depends on the ODE family: some parameters are uniquely identifiable, while in other families only invariant parameter combinations are identifiable or external physical calibration is required. For laws that are at most linear in velocity, compatibility forces affine coordinate alignment, yielding explicit parameter relations, invariants, and calibration conditions. This affine conclusion extends to broader finite velocity-feature families when coordinate curvature can be separated from the declared velocity dependence. For families admitting a squared-velocity term, nonlinear coordinate ambiguity can remain; a law-derived normalization instead enables affine comparison between canonical laws. Experiments on synthetic systems and real pendulum and free-fall videos support the predicted parameter relations, coverage effects, and calibration requirements.

Figures & tables

Declared family	Coordinate gauge	Invariant information	Sufficient parameter calibration
LTI $z^{\prime\prime}+\delta z^{\prime}+\alpha z=0$	$\lambda u+\tau$ , $\alpha\tau=0$	$\delta,\alpha$	None
Angular pendulum $z^{\prime\prime}+\delta z^{\prime}+\omega^{2}\sin z=0$ ( $\omega>0$ )	$\sigma u+2\pi k$	$\delta,\omega$	None
Normalized Van der Pol $z^{\prime\prime}+\mu(z^{2}-1)z^{\prime}+z=0$ ( $\mu\neq 0$ )	$\sigma u$	$\mu$	None
Cubic Duffing $z^{\prime\prime}+\delta z^{\prime}+\alpha z+\beta z^{3}=0$ ( $\beta\neq 0$ )	$\lambda u$	$\delta,\alpha,$ $\operatorname{sign}(\beta)$	One unsigned amplitude anchor for $\beta$
Quintic Duffing $\begin{gathered}z^{\prime\prime}+\delta z^{\prime}+\alpha z\\[-1.00006pt] {}+\beta z^{3}+\gamma z^{5}=0\end{gathered}$ ( $\beta\gamma\neq 0$ )	$\lambda u$	$\delta,\alpha,$ $\operatorname{sign}(\beta),$ $\gamma/\beta^{2}$	One unsigned amplitude anchor for $\beta,\gamma$

Table 1: Family-specific coordinate ambiguity, identifiable parameter information, and sufficient physical calibration under the assumptions of Theorem 3.1 . Details are given in Appendix J .

Figure 1: Coordinate maps explain parameter orbits. Top: invariant ratios and coefficient orbits; bottom: learned coordinates and allowed affine maps. Colors match across rows; D–F reuse training-fitted scales for testing.

Velocity design	Rank	Coordinate $e_{\rm map}$ (%)	Parameter $e_{\theta}$ (%)
One-way, $K=6$	$3/4$	1.18 [1.08, 1.40]	44.97 [15.07, 59.87]
Two-way, $K=1$	$2/4$	1.08 [0.91, 1.34]	6.95 [4.85, 8.65]
Two-way, $K=2$	$4/4$	1.07 [0.93, 1.22]	1.34 [0.85, 2.60]

Table 2: Accurate coordinates do not ensure accurate drag coefficients. Ten IC collections

\times

five optimizer seeds per condition. Rank is the minimum for

[1,v,v|v|,v^{2}]

over shared positions.

K

counts speed magnitudes;

e_{\theta}

tests the scale-corrected mean drag coefficient.

Raw coordinate error $e_{\rm map}$ (%)	Canonical coordinate error $e_{\rm map}$ (%)	Restoring-force error $e_{\rm dyn}$ (%)
5.633 [5.629, 5.635]	0.305 [0.303, 0.305]	0.193 [0.183, 0.199]

Table 3: Law-derived normalization. Median [Q25,Q75] over three fits.

Experiment	Parameter	Reference	Estimate	$e_{\theta}$ (%)	$e_{\rm map}$ (%)
Pendulum (nonlinear)	$L$	0.50	$0.5150\pm 0.0012$	2.92 [2.82, 3.17]	5.04 [4.38, 5.83]
Pendulum (LTI)	$L_{\rm eff}$	0.50	$0.5646\pm 0.0005$	12.92 [12.83, 12.97]	7.13 [5.46, 7.45]
Side fall	$g$	9.81	$9.370\pm 0.771$	5.10 [3.21, 7.42]	3.75 [3.22, 5.14]
Overhead fall	$g$	9.81	$10.146\pm 0.029$	3.35 [3.26, 3.39]	5.57 [5.57, 5.60]

Table 4: Real-video coordinate and physical parameter recovery. Estimates are mean

\pm

SD; errors are median [Q25,Q75].

L

is in m and

g

\mathrm{m/s^{2}}

Appendix figures & tables38 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Meaning
$\mathcal{D}$	Observed collection of timestamped video frames.
$m,M,I^{(m)},T_{m}$	Clip index, number of clips, underlying time interval, and final frame index; clip $m$ has $T_{m}+1$ frames.
$t_{m,k},\Delta t_{m}$	Recorded timestamp and known uniform within-clip sampling interval. All dynamical derivatives use this time coordinate.
$x^{(m)}(t),x_{k}^{(m)}$	Ideal frame path and its recorded sample at $t_{m,k}$ .
$z^{(m)}(t)$	Unobserved data-generating scalar physical coordinate.
$E_{\phi}$	Shared per-frame scalar encoder. Its output enters the ODE residual directly.

Appendix

Table 5: Notation index.

Panel of Figure 2 — Figure 2: From joint video fitting to parameter recovery. The declared family and coverage conditions determine the raw or canonical route. In the canonical route, $\psi_{\eta}(u_{\star})=0$ and $\psi_{\eta}^{\prime}(u_{\star})=1$ fix a normalizer, and $r^{\prime\prime}=\psi_{\eta}^{\prime}F_{\eta}+\psi_{\eta}^{\prime\prime}(\hat{z}^{\prime})^{2}$ is the transformed law. Normalization is computed from the fitted law; external anchors are added only when the desired target is not yet identified. Parameter sets are understood with the containment and realizability qualifications in Appendix D . Synthetic coordinate and parameter errors evaluate these predictions using reference states. The diagram uses $z$ for physical state and $\hat{z}$ for learned state; experimental plots use $q$ and $z$ , respectively.

Question	Evidence to report	Scope of the conclusion
Declared family	Coefficient restrictions and domains; test-law error and matched-family comparisons.	A small fitting residual alone does not establish that the declared family contains the physical law.
Shared state map	One frozen map across clips; held-out coordinate error and, when available, derivative comparisons.	Coordinate agreement supports transfer of the map. A shared encoder alone does not establish latent-state consistency.
Velocity coverage	Physical coordinate, level tolerance, distinct speeds and signs, feature rank and conditioning.	The coefficient-matching result requires its stated coverage. Failure of a sufficient rank condition does not itself prove nonidentifiability.
Canonical normalization	Fitted $c_{2}$ , normalizer basepoint, regular interval, and transformed-law comparison.	The comparison is restricted to the valid chart. Normalization removes the squared-velocity channel but does not determine physical units.
Parameter recovery	Allowed coordinate action and branches, parameter-orbit error, and the specified invariants.	Agreement of selected coefficients or invariants need not establish full law equivalence or identify every parameter.
Physical calibration	Anchor values, units, domain, remaining branches, and a uniqueness argument.	Calibrated estimates use additional physical information. A regular full-rank anchor map gives local uniqueness, with branches checked separately.

Appendix

Table 6: Empirical checks and the conclusions they support. The checks probe the stated premises; finite observations do not certify their continuous-time versions.

Family and required design	Coordinate relation	Parameter conclusion
Semilinear; three distinct physical velocities at each covered state (Theorem 3.1 )	Raw coordinates are affine-related	Affine compatible outer set; invariant combinations, singleton identification, and sufficient local physical anchors
Finite dilation-stable library with constants, excluding $v^{2}$ ; full augmented rank (Theorem 4.1 )	Raw coordinates are affine-related	Affine compatibility analysis, restricted to the declared family
Polynomial in velocity, admitting $c_{2}(z)v^{2}$ ; $d_{v}+1$ distinct physical velocities per state (Theorem 4.2 )	Raw relation may be nonlinear; canonical relation is affine	Canonical compatible outer set in the original parameter space; invariants and local physical-anchor analysis
Required coverage or shared-map premise fails or remains unverified	The affected guarantee is unavailable	Acquire additional evidence, supply a separate family-specific argument, or withhold the affected claim

Appendix

Table 7: Parameter-recovery routes within a fixed declared family, conditional on a shared non-collapsed state map, exact compatibility, and the stated domain and coverage assumptions. The semilinear row is a special case of the finite-feature route; the rows are not an exhaustive partition of scalar ODEs.

Figure 3: Temporal evolution over two reference oscillation times. Nine observations at $0,T/4,\ldots,2T$ show amplitude and timing changes. Each row starts at a turning point; $T$ is the first interval between same-type turning points, not an assumed constant period. Damped systems ring down, whereas Van der Pol approaches a limit cycle. All frames are $64\times 64$ .

Family / experiment	Pixels / Hz / duration	Train / test
Affine LTI	$64^{2}$ / 60 / 6 s	16 / 8
Pendulum	$64^{2}$ / 60 / 8 s	10 / 4
Van der Pol	$64^{2}$ / 60 / 12 s	6 / 4
Cubic Duffing	$64^{2}$ / 60 / 8 s	10 / 4
Quintic Duffing	$64^{2}$ / 60 / 3 s	16 / 8
Quadratic / Helmholtz	$64^{2}$ / 60 / 8 s	3 / 4

Appendix

Table 8: Synthetic observations and data splits. Clip counts are per IC collection. Each family uses ten collections

\times

ten optimizer seeds. Training clips supply images and timestamps for fitting the encoder and law. Test clips supply the reported held-out errors (coordinate NRMSE

e_{\rm map}

(%)). The permitted coordinate comparison is fitted using training reference pairs and frozen on test clips.

Family	Training pairs	Test pairs
LTI / quintic	$(\pm a,0)$ , $a\in\{0.25,0.55,0.85,1.15\}$ ; $(0,\pm b)$ , $b\in\{0.8,1.8,2.8,3.4\}$	$(\pm a,0)$ , $a\in\{0.65,1.00\}$ ; $(0,\pm b)$ , $b\in\{0.40,1.00\}$
Pendulum	$(\pm\pi/2,\pi h)$ , $h\in\{-1/2,-1/4,0,1/4,1/2\}$	$(\pm\pi/2,\pm\pi/8)$
Van der Pol	$(0,\pm 0.5)$ , $(0,\pm 2.5)$ , $(\pm 3,0)$	$(0,\pm 1.25)$ , $(\pm 2.75,0)$
Cubic	$(0,\pm\sqrt{4p^{2}+2p^{4}})$ , $p\in\{0.40,0.55,0.70,0.85,1.00\}$	$(0,\pm\sqrt{4p^{2}+2p^{4}})$ , $p\in\{0.625,0.925\}$
Quadratic	$(0,-1.2)$ , $(0,0.8)$ , $(0,1.2)$	$(-0.4,0.65)$ , $(0.4,-0.65)$ , $(-0.15,1.0)$ , $(0.15,-1.0)$

Appendix

Table 9: Nominal initial states before IC perturbation. Pairs are

(q_{0},v_{0})

in the physical coordinate. Each

\pm

includes both signs; in the pendulum row, angle and velocity signs vary independently. LTI and quintic use the same IC design.

Learned law / physical coefficients	Allowed map / parameter consequence
$z^{\prime\prime}+\hat{\delta}z^{\prime}+\hat{\alpha}(z-\hat{c})=0$ $(\delta,\alpha,c)=(0.2,4,0)$	$z\simeq\lambda q+\tau$ ; $\hat{c}\simeq\lambda c+\tau$ ; $\delta,\alpha$ invariant.
$z^{\prime\prime}+\hat{\delta}z^{\prime}+\hat{\omega}^{2}\sin z=0$ $(\delta,\omega^{2})=(0.15,4)$	$z\simeq sq+2\pi n$ ; $s=\pm 1$ , $n\in\mathbb{Z}$ ; $\delta,\omega^{2}$ invariant.
$z^{\prime\prime}-\hat{\mu}(1-z^{2})z^{\prime}+z=0$ $\mu=1.5$	$z\simeq\pm q$ ; $\mu$ invariant.
$z^{\prime\prime}+\hat{\delta}z^{\prime}+\hat{\alpha}z+\hat{\beta}z^{3}=0$ $(\delta,\alpha,\beta)=(0.2,4,4)$	$z\simeq\lambda q$ ; $\hat{\beta}\simeq\beta/\lambda^{2}$ ; $\delta,\alpha,\operatorname{sign}\beta$ invariant.
$z^{\prime\prime}+\hat{\delta}z^{\prime}+\hat{\alpha}z+\hat{\beta}z^{3}+\hat{\gamma}z^{5}=0$ $(\delta,\alpha,\beta,\gamma)=(0.2,4,3,2)$	$z\simeq\lambda q$ ; $\hat{\gamma}\simeq\gamma/\lambda^{4}$ ; also $\gamma/\beta^{2}$ invariant.
$z^{\prime\prime}+\hat{\delta}z^{\prime}+\hat{\alpha}z+\hat{\beta}z^{2}=0$ $(\delta,\alpha,\beta)=(0.18,1.6,0.8)$	Origin: $z\simeq\lambda q$ , $\hat{\alpha}\simeq\alpha$ , $\hat{\beta}\simeq\beta/\lambda$ . Shifted: $z\simeq\lambda(q+\alpha/\beta)$ , $\hat{\alpha}\simeq-\alpha$ .

Appendix

Table 10: Declared laws, fixed physical coefficients and permitted coordinate actions. Nonzero

\lambda

is estimated from training coordinate pairs. The quintic invariant uses

\beta\neq 0

; the plotted physical coefficients are positive.

Family	IC collections	Seeds per set	Updates	$\mathrm{lr}_{\mathrm{enc}}/\mathrm{lr}_{\mathrm{ode}}$
Affine LTI	10	10	2,000	$0.0005/0.003$
Pendulum	10	10	2,500	$0.001/0.005$
Van der Pol	10	10	5,000	$0.001/0.003$
Cubic Duffing	10	10	8,000	$0.0005/0.1$
Quintic Duffing	10	10	8,000	$0.0005/0.003$
Quadratic / Helmholtz	10	10	6,000	$0.0005/0.003$

Appendix

Table 11: Training settings for the six-family experiment. All use Adam and the final checkpoint after the stated number of updates (optimizer steps). Learning rates are encoder / dynamical law. Each family has 100 fits.

Family (100 fits)	Parameter $e_{\theta}$ (%)	Coordinate $e_{\rm map}$ (%)
Affine LTI	0.85 [0.52, 1.65]	1.68 [1.53, 1.90]
Pendulum	1.44 [0.84, 2.17]	3.58 [2.27, 5.99]
Van der Pol	0.31 [0.13, 0.61]	2.45 [1.69, 3.71]
Cubic	2.56 [1.60, 4.60]	0.82 [0.68, 1.08]
Quintic	1.68 [1.02, 2.29]	0.78 [0.62, 1.06]
Quadratic	2.47 [1.31, 4.37]	3.29 [1.96, 6.08]

Appendix

Table 12: Parameter relative error and coordinate recovery. Percentages are median [Q25, Q75] over all 100 fits per family.

Figure 4: Test coordinate NRMSE

e_{\rm map}

by IC collection. Each family contains ten IC collections, each with ten optimizer seeds. Dots show

e_{\rm map}

; black bars give each collection’s median and interquartile range. Colors match Figure 1 ; IC indices are local to each family. The logarithmic vertical limits vary by family.

Figure 5: Parameter relative error

e_{\theta}

by IC collection. The same fits and IC groups as in Figure 4 , now showing

e_{\theta}

. Within each group, horizontal offsets follow optimizer-seed order and are identical in both figures. Bars show median and interquartile range.

Figure 6: Matched affine-LTI comparison across IC collections. Each family has ten IC collections and ten paired optimizer seeds per collection. Faint lines show individual seeds; solid curves and bands give the within-IC median and interquartile range of test dynamics relative RMSE

e_{\rm dyn}

(%), defined in Eq. ( 57 ) and evaluated on physical accelerations. Dashed curves show the training-fitted physical-state LTI oracle. Both methods in A use the affine-LTI family.

Dataset	Affine LTI (%)	Family model (%)	Family wins
Affine LTI	3.86 [1.96, 6.58]	4.56 [2.16, 6.84]	46/100
Pendulum	11.34 [11.20, 11.70]	2.17 [1.43, 3.89]	99/100
Van der Pol	88.48 [88.01, 88.79]	5.80 [3.07, 9.21]	98/100
Cubic	17.92 [17.46, 18.53]	0.81 [0.57, 1.39]	100/100
Quintic	24.29 [23.46, 25.02]	0.54 [0.33, 0.89]	100/100
Quadratic	22.03 [20.42, 23.85]	3.15 [1.70, 6.68]	92/100

Appendix

Table 13: Test dynamics relative RMSE

e_{\rm dyn}

(%). Physical-acceleration comparison using Eq. ( 57 ); median [Q25, Q75] over 100 pairs per family. Wins count pairs with a smaller

e_{\rm dyn}

for the declared family model.

Figure 7: Opposite odd-drag crossings. Both rows show $96\times 96$ inputs at 60 Hz, aligned at the same position at $t=0$ . The central frames have the same geometry despite opposite velocities; labels give time in seconds.

IC collections	Seeds per collection	Updates	$\mathrm{lr}_{\mathrm{enc}}/\mathrm{lr}_{\mathrm{ode}}$
10	5	5,000	$0.0005/0.002$

Appendix

Table 14: Coverage training settings. Adam; each coverage condition uses ten IC collections and five optimizer seeds per collection.

Figure 8: Velocity coverage, parameter recovery, and coordinate accuracy. A: minimum rank of

[1,v,v|v|,v^{2}]

over 21 shared positions in

[-0.2,0.2]

; all ten IC collections give the displayed rank for each condition. The dashed line marks full rank. B: coordinate NRMSE

e_{\rm map}

(Eq. ( 55 )) on the same four test clips within each IC collection. C: parameter relative error

e_{\theta}

of the scale-corrected mean drag coefficient (Eq. ( 56 )). B/C show all 50 fits per condition, colored by IC collection, with matching horizontal offsets. Black bars show the median and interquartile range over the 50 fits, using the convention of Figures 4 and 5 ; numbers give medians (%). All axes are linear.

K

counts speed magnitudes.

Panel of Figure 9 — Figure 9: Projected pendulum observations. An orthographic camera looks along the vertical axis, retaining $X=L\sin q$ and discarding the depth coordinate $-L\cos q$ . The projected rod and constant-size bob produce the image coordinate $y=\sin q$ . Bottom: nine $96\times 96$ observations from $0$ to $3T/2$ for a $70^{\circ}$ release; $T$ is the first full oscillation duration. The illustrative strip continues the original ODE and renderer beyond the 4.8 s training-clip duration.

Damping $e_{\theta,\delta}$	Restoring coefficient $e_{\theta,\omega^{2}}$	Quadratic channel $e_{\theta,\rho}$
0.64% [0.53%, 0.68%]	0.21% [0.20%, 0.23%]	1.93% [1.83%, 1.95%]

Appendix

Table 15: Fitted-law coefficient errors. Parameter relative errors

e_{\theta,j}

against

(\delta,\omega^{2},\rho)=(0.15,4,1)

; median [Q25,Q75] over three fits on the same IC collection.

Figure 10: Coordinate and force recovery across three fits. A: held-out angle residuals after training-fitted affine alignment. B: canonical force residual

(\widetilde{F}-F)/\mathrm{RMS}(F)

in percent, with

F=-4\sin q

. Thick curves: pointwise medians; shading: Q25–Q75; thin curves: individual fits. Coordinates are binned for display only;

e_{\rm map}

uses original test frames.

Training observations	Videos	Distinct velocities	Raw $e_{\rm map}$ (%)	Canonical $e_{\rm map}$ (%)	Force $e_{\rm dyn}$ (%)
$60^{\circ}$ ; 1 passage	1	1	3.137 [3.092,3.140]	0.898 [0.886,0.899]	19.21 [19.19,20.02]
$60^{\circ}$ ; 2 passages	1	2	8.441 [8.434,8.725]	0.823 [0.560,1.192]	3.49 [3.38,4.61]
$60^{\circ}$ ; 3 passages	1	3	8.129 [8.114,8.196]	0.691 [0.657,0.719]	2.65 [1.91,3.48]
$50^{\circ},60^{\circ},70^{\circ}$	3	3	2.853 [2.839,2.943]	0.748 [0.672,0.762]	9.55 [8.59,10.05]
$(50^{\circ},60^{\circ},70^{\circ})\times 2$	6	3	3.007 [2.956,3.038]	0.539 [0.527,0.546]	3.83 [3.22,5.43]

Appendix

Table 16: Trajectory coverage and repeated observations. Errors are median [Q25,Q75] over five fits per row; all 25 maps are defined. Three-angle rows use one passage per video. Velocity counts give the number of distinct velocities per state on

U=[-0.4,0.4]

rad.

\times 2

denotes two renderings per angle with unchanged trajectories.

Figure 11: Falling-object recording settings in IRIS and our collection. IRIS examples are from Khanbayov et al. (2026) . Left: side view; right: overhead view. Each half contains three IRIS settings above three self-captured release heights, with one recorded frame per setting. IRIS side-view examples vary the ball and framing; the overhead examples show different viewpoints and framing. Our examples use the same ball and camera configuration within each view. Images preserve aspect ratio; self-captured images use a common fixed crop within each view. These are acquisition examples, not matched physical states or performance comparisons.

Figure 12: Pendulum observations on IRIS ( Khanbayov et al., 2026 ) . Each row shows recording 01 from one release-angle setting. The displayed RGB frames illustrate the observed motion; elapsed time is measured from the first selected frame.

Model	Affine: $z\simeq\lambda q+\tau$	Sign/period: $z\simeq sq+2\pi n$
Nonlinear pendulum	5.04 [4.38, 5.83]	13.22 [12.20, 13.97]
LTI	7.13 [5.46, 7.45]	39.04 [28.65, 53.30]

Appendix

Table 17: Coordinate accuracy under two common alignment rules. Coordinate NRMSE

e_{\rm map}

from ( 55 ) (%), median [Q25, Q75] over 25 paired fits per model. Both alignments are fitted on training recordings and evaluated on the same 27 held-out recordings per pair.

Figure 13: Coordinate recovery and calibrated length by training group. Groups 01–05 each contain five nonlinear fits (teal circles) and five matched LTI fits (orange triangles). A–B: test coordinate NRMSE

e_{\rm map}

from ( 55 ) after affine and sign/period alignment, respectively; bars show median and IQR. C: length after gravity calibration in ( 62 ); bars show mean

\pm

sample SD, and the dashed line marks the IRIS reference

0.50

m. All vertical axes are linear.

Figure 14: Side-view observations. Three release heights (0.8, 1.0, and 1.2 m), one row each. The fixed camera captures an intermediate vertical interval; the release instant is unavailable in these windows. The first seven frames of recording 01 are shown from each selected window; shared elapsed times are relative to the start of each window. Fixed-crop RGB images are $72\times 192$ .

Figure 15: Side-view physical calibration. The measured ball width and diameter reference define the spatial conversion used to report gravity.

Figure 16: Affine parameter orbit, coordinate recovery, and calibrated gravity. Colors identify the five training IC collections, IC1–IC5; each point is one of five fits per collection. A: learned acceleration

A

versus coordinate scale

\lambda_{\rm px}

; the dashed line is the theoretical relation

A=\lambda_{\rm px}a_{\rm px}

from ( 63 ), using the independently fitted pixel-acceleration reference. B: held-out coordinate NRMSE

e_{\rm map}

from ( 55 ); black bars show median and IQR. C: gravity after diameter calibration in ( 64 ); black bars show mean

\pm

sample SD, and the dashed line marks

9.81\,\mathrm{m/s^{2}}

Figure 17: Overhead observations. Fixed-crop RGB frames at release heights 1.0, 1.2, and 1.4 m, one row each. Times are relative to the start of the corresponding original clip.

Collection	0.8 m	1.0 m	1.2 m	1.4 m	Videos	Frames
IC1	01	01	01,02	01	5	95
IC2	01	01,02	01,02,03	01	7	132
IC3	01,02	01,02	01,02,03	01,02	9	166
IC4	01,02,03	01,02,03	01,02,03	01,02	11	199
IC5	01,02,03,04	01,02,03,04	01,02,03,04	01,02	14	247
Test	05	05	05	03,04,05	6	78

Appendix

Table 18: Overhead training and test recordings. IC1–IC5 are nested training collections of repeated releases; each covers all four heights and shares the same six test videos. Recording IDs are local to each release height. Frames are counted within the training segments or test windows actually used.

Raw $e_{\rm map}$ (%)	Canonical $e_{\rm map}$ (%)	Learned $\rho$
15.85 [15.74, 16.01]	5.57 [5.57, 5.60]	$2.072\pm 0.024$

Appendix

Table 19: Law-derived normalization in overhead fall. IC5 with

\rho_{0}=2.0

, five fits. Coordinate errors

e_{\rm map}

use ( 55 ) and are median [Q25, Q75];

\rho

is mean

\pm

sample SD.

Figure 18: Coordinate residuals for IC5. All five fits use

\rho_{0}=2.0

. Test-coordinate residuals after the training-fitted raw or canonical affine readout are averaged in twelve inverse-radius bins. Thin curves show individual fits; thick curves show pointwise medians. Zero indicates agreement with the inverse-radius reference. Physical anchors are not used in these readouts.

Figure 19: Release-height anchors. Original RGB frame zero from seven recordings, shown before masked-saturation preprocessing. Labels give nominal release height and recording ID in Table 18 . Calibration applies the same preprocessing and frozen encoder used for the dynamic observations.

Figure 20: Sensitivity to the training IC collection. IC1–IC5 use the trajectory collections in Table 18 , with

\rho_{0}=2.0

and five paired local network initializations. A: paired raw/canonical test NRMSE on the same six test videos; horizontal bars and labels give medians; vertical bars show the IQR. B: learned

\rho

, with reference 2. C: gravity from release-height calibration, with reference

9.81\,\mathrm{m/s^{2}}

. Bars and labels in B/C show mean

\pm

sample SD.

IC collection	$\rho_{0}$	Learned $\rho$	$\hat{g}$ ( $\mathrm{m/s^{2}}$ )	Raw	Canonical
IC1	1.0	$2.037\pm 0.012$	$9.037\pm 0.040$	12.81 [12.78, 12.83]	3.86 [3.70, 3.88]
IC1	2.0	$2.160\pm 0.026$	$8.842\pm 0.032$	13.35 [13.13, 13.35]	3.84 [3.78, 3.90]
IC1	3.0	$2.233\pm 0.079$	$8.674\pm 0.054$	13.42 [13.29, 13.70]	3.85 [3.82, 3.94]
IC2	1.0	$2.040\pm 0.025$	$9.380\pm 0.128$	12.58 [12.57, 12.80]	4.17 [4.15, 4.24]
IC2	2.0	$2.091\pm 0.047$	$9.245\pm 0.109$	12.87 [12.68, 12.91]	4.19 [4.17, 4.28]
IC2	3.0	$2.196\pm 0.095$	$8.960\pm 0.067$	12.95 [12.84, 13.29]	4.26 [4.25, 4.39]

Appendix

Table 20: IC collections and coefficient initialization in overhead fall. Five fits per row, 75 fits in total. Coefficient and gravity estimates are mean

\pm

sample SD; coordinate errors are median [Q25, Q75] NRMSE (%). IC1–IC5 are defined in Table 18 . All gravity estimates use release-height calibration. Settings share six test clips and seven calibration frames. The main experiment uses IC5 (14 videos) and

\rho_{0}=2.0

Figure 21: Cubic Duffing example in the supplementary tool. For the displayed nonzero- $\beta$ family, the tool reports $\alpha$ and $\delta$ as fixed across the analyzed coordinate orbit, while $\beta$ varies with coordinate scale.

Figure 22: Velocity coverage for quadratic drag. The tool lists the required rank-four feature set $[1,v,v|v|,v^{2}]$ at each physical state and an algebraically feasible velocity example $\{1,-1,2,-2\}$ .

Figure 23: Canonical route for overhead fall. For $z^{\prime\prime}=\rho(z^{\prime})^{2}/z-kz^{2}$ , the tool directs the comparison to the canonical coordinate before assessing parameter compatibility.

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Mohamed bin Zayed University of Artificial Intelligence, Abu Dhabi, UAE · University of Melbourne, Melbourne, Australia · Carnegie Mellon University, Pittsburgh, USA
Limits of Learning Linear Dynamics from Experiments
May 12, 2026Aybüke Ulusarslan, Niki Kilbertus, Nora SchneiderSystem Identification Identifiability
1Technical University of Munich · 2Helmholtz Munich · 3Munich Center for Machine Learning (MCML)
Identifiability Guarantees for Drivers and Dynamics of Delayed Physical Systems
Sep 29, 2026Julien Boussard, Antoine Debouchage, Théo SaulusIdentifiability Ordinary Differential Equations
School of Computer Science, McGill University, Canada · Mila - Quebec AI Institute, Canada · LaMMe, Université Évry Paris-Saclay, France +1

May 27, 2026, cs.CVJ/K move · Enter open · S save

Physics from Video: Identifiability of Time-Invariant Second-Order ODEs under Minimal Trajectory Conditions

Yuanyuan Wang, Wenjie Wang, Kun Zhang, Mingming Gong

Mohamed bin Zayed University of Artificial Intelligence, Abu Dhabi, UAE · University of Melbourne, Melbourne, Australia · Carnegie Mellon University, Pittsburgh, USA

Identifiability Ordinary Differential Equations Collapse

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Bridging the gap between visual realism and physical understanding is a core challenge for video-based world models. We study the structural identifiability of continuous-time physical laws from raw pixels, focusing on whether an encoder-only pipeline can uniquely recover the parameters of second-order linear ODEs. We prove that a level-set slope-coverage condition ensures the learned latent space is locally affine to the true physical state, enabling exact parameter recovery. Our theory provides the first characterization of minimal data requirements across damping regimes, establishing that underdamped systems are identifiable from a single video clip, whereas other regimes require three diverse trajectories. We further introduce a variance-floor regularizer to stabilize the decoder-free objective and prevent latent collapse. Validated on synthetic and real-world data, our approach demonstrates that interpretable physical constants can be reliably estimated from video without the need for compute-intensive pixel reconstruction, ensuring both physical correctness and transparency. Code is available at https://github.com/wenjiewang3/PhysicsFromVideo.

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