Neural ODEs Meet Concurrent Learning: Stable Online Learning with Lyapunov Guarantees

Omkar Sudhir Patil

eess.SYSep 26, 2026

Neural ODEs Meet Concurrent Learning: Stable Online Learning with Lyapunov Guarantees

Authors: Omkar Sudhir Patil

Organizations: Division of Electrical and Computer Engineering Louisiana State University Baton Rouge, LA 70803, USA

Abstract

Neural ODEs learn dynamics from trajectory losses, but their adjoint gradients lack the regressor-times-parameter-error structure on which Lyapunov analyses of online adaptation rest, so training on streaming data comes without stability guarantees. We show that this structure is in fact present: the adjoint gradient decomposes exactly into a positive semi-definite trajectory operator acting on the parameter error plus a nonlinear perturbation with explicit, horizon-dependent bounds. A quadratic Lyapunov function then certifies online Neural ODE training over sliding windows under computable gain and horizon conditions, and the same certificate extends to stored data: its drift branch recovers concurrent learning, and its trajectory branch yields NODE-CL, a stored-segment Gauss-Newton method built on batched forward sensitivities that needs no state-derivative estimates. On four DeepMind Control Suite domains, NODE-CL attains the lowest median prediction error on three under velocity measurement noise, where observer-based concurrent learning degrades by up to 8x; with clean measurements it is best on the pendulum and within a factor of 1.6 of the best stored-data baseline on the cartpole and reacher.

Figures & tables

Figure 1: Streaming data (01) feed two residual pathways (02), an instantaneous drift residual formed with an observer and a trajectory residual formed by rolling the model out from stored initial states; each gradient decomposes into a positive semi-definite operator on the parameter error plus a bounded perturbation, so both pathways share one certified update (03).

	pendulum ( $p=73$ )		cartpole ( $p=146$ )		acrobot ( $p=162$ )		reacher ( $p=162$ )
Method	c	n	c	n	c	n	c	n
single-step	$1.643$ (0)	$1.650$ (0)	$3.287$ (0)	$3.145$ (0)	$1.621$ (1)	$\mathbf{1.232}$ (2)	$1.509$ (0)	$1.516$ (0)
CL (constant gain)	$0.315$ (0)	$0.609$ (0)	$0.623$ (0)	$1.363$ (0)	$3.615$ (0)	$4.385$ (0)	$1.007$ (0)	$0.976$ (0)
CL-LS ( Hart et al., 2025a )	$0.195$ (0)	$1.642$ (0)	$\mathbf{0.251}$ (4)	$1.984$ (0)	$1.409$ (0)	$2.015$ (0)	$\mathbf{0.064}$ (2)	$0.215$ (0)
NODE-replay (control)	$0.183$ (0)	$0.141$ (3)	$0.311$ (0)	$0.741$ (1)	$\mathbf{1.002}$ (2)	$2.129$ (1)	$0.077$ (2)	$0.088$ (2)
NODE-CL (proposed)	$\mathbf{0.035}$ (5)	$\mathbf{0.038}$ (2)	$0.381$ (1)	$\mathbf{0.302}$ (4)	$1.545$ (2)	$1.279$ (2)	$0.069$ (1)	$\mathbf{0.075}$ (3)

Table 1: Benchmark suite (five evaluation seeds, tuning seed excluded,

60\,

s): held-out

1\,

s prediction error normalized by the state standard deviation, median over seeds, clean (c) and with velocity noise

\sigma_{v}=3\times 10^{-3}

(n); in parentheses, seeds on which it is best; best median in bold. Initial-model errors:

2.16

2.08

1.52

1.71

. CL-LS is the identification form of the DNN concurrent-learning law of Hart et al. (2025a) (stored-set extension of Proposition 2 , Appendix J ); CL is its constant-gain simplification; NODE-CL is the proposed method in Theorem 2 .

Figure 2: Held-out

1\,

s error against velocity noise per domain (five seeds, median and range; dotted, initial model). The

10^{-2}

level was not run for the acrobot and reacher.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: Normalized functional error over time (three seeds, geometric mean with seed range) for (a) S1 and (b) S2.

Method	func. error [range]	$\left\\|\tilde{\theta}(t_{f})\right\\|/\left\\|\tilde{\theta}(0)\right\\|$	func. error [range]	pred. ( $4\,$ s)	S1	S2
	S1 (teacher, $p=25$ )		S2 (manipulator, $p=86$ )		cost (ms)
single-step	$6.8$ e $-2$ [ $3.1$ e $-2$ , $1.1$ e $-1$ ]	$0.89$	$3.2$ e $-1$ [ $2.2$ e $-1$ , $5.1$ e $-1$ ]	$0.293$	$0.03$	$0.04$
drift window (Alg. 1)	$9.3$ e $-3$ [ $3.7$ e $-3$ , $1.7$ e $-2$ ]	$0.80$	$7.8$ e $-2$ [ $6.1$ e $-2$ , $1.1$ e $-1$ ]	$0.119$	$0.15$	$0.81$
adjoint window (Alg. 2)	$2.9$ e $-2$ [ $6.8$ e $-3$ , $7.3$ e $-2$ ]	$0.85$	$8.9$ e $-2$ [ $8.0$ e $-2$ , $9.9$ e $-2$ ]	$0.122$	$1.4$	$2.5$
online Adam (adjoint)	$2.4$ e $-2$ [ $9.0$ e $-3$ , $7.7$ e $-2$ ]	$1.31$	$1.2$ e $-1$ [ $9.4$ e $-2$ , $1.8$ e $-1$ ]	$0.157$	$1.4$	$2.5$
CL, $k_{f}=200$	$6.9$ e $-3$ [ $2.6$ e $-3$ , $1.2$ e $-2$ ]	$0.83$	$2.3$ e $-2$ [ $2.1$ e $-2$ , $2.6$ e $-2$ ]	$0.0375$	$0.06$	$0.14$

Appendix

Table 2: Diagnostic studies S1 and S2 (three seeds,

120\,

s; these runs use the scalar bounded-gain forgetting of Slotine and Li (1989) in the least-squares laws): terminal normalized functional error (geometric mean, seed range); final parameter error ratio (S1); held-out

4\,

s prediction error (S2; initially

14.3

); update cost in ms on one CPU core, excluding stack selection. All rows except online Adam are sampled implementations of continuous-time laws covered by the analysis under its stated assumptions and gain conditions.

Figure 4: Terminal normalized functional error after

60\,

s against the standard deviation of white velocity measurement noise (three seeds, geometric mean with seed range) for (a) S1 and (b) S2.

Domain	Method	func (c)	$1\,$ s (c)	$4\,$ s (c)	func (n)	$1\,$ s (n)	$4\,$ s (n)	upd ms	sel ms/s
pendulum	single-step	$1.0e+00$	$1.643$ [1.00, 2.05]	$1.27$	$8.7e-01$	$1.650$	$1.24$	$0.03$	$0$
pendulum	CL	$2.5e-02$	$0.315$ [0.17, 0.73]	$0.29$	$1.3e-01$	$0.609$	$0.70$	$0.09$	$44$
pendulum	CL-LS	$2.3e-02$	$0.195$ [0.03, 0.25]	$0.11$	$2.0e-01$	$1.642$	$0.90$	$0.49$	$40$
pendulum	NODE-replay	$5.0e-03$	$0.183$ [0.06, 0.26]	$0.12$	$1.5e-02$	$0.141$	$0.11$	$4.77$	$188$
pendulum	NODE-CL	$2.7e-03$	$0.035$ [0.01, 0.04]	$0.02$	$1.2e-02$	$0.038$	$0.03$	$5.32$	$186$
cartpole	single-step	$4.6e-01$	$3.287$ [1.65, 4.44]	$7.50$	$4.5e-01$	$3.145$	$8.28$	$0.04$	$0$

Appendix

Table 3: Benchmark suite, full results (five evaluation seeds, medians,

60\,

s): terminal normalized functional error on visited states, held-out

1\,

s error with seed range, held-out

4\,

s error, for clean (c) and noisy (n,

\sigma_{v}=3\times 10^{-3}

) measurements; mean cost per update and amortized stack-selection cost per simulated second (ms, one CPU core, NumPy). The

10^{-3}

and

10^{-2}

levels are shown in Figure 2 ; the

10^{-2}

level was not run for the acrobot and reacher.

Figure 5: Benchmark suite learning curves: normalized functional error on the visited states against streaming time (five evaluation seeds, median with seed range, clean measurements).

Figure 6: Held-out

1\,

s prediction error per domain (five evaluation seeds, median with seed range), clean (solid) and under velocity noise

3\times 10^{-3}

(hatched); the dotted line is the initial model. Learning curves are given in Figure 5 .

Figure 7: Additional ablations (three seeds,

60\,

s; geometric means; violation rates and excited dimensions are arithmetic means, and zero violations are drawn as zero). (a) Gain-matrix variants at

\alpha=2

: terminal functional error and the smallest eigenvalue of

\Gamma

attained during the run. (b) Excitation richness: terminal functional error and the dimension of the excited subspace against the number of multisine components at equal total power. (c) Projection metric on a tight parameter set (

\bar{\theta}=\left\|\theta_{\mathrm{tr}}\right\|

\epsilon_{\theta}=0.3

): terminal functional error and the fraction of updates that violate Equation 11 . (d) Sigma-modification gain: functional and parameter error. (e) Network width in S2 for the drift and adjoint laws.

Figure 8: S1, five seeds, windowed laws and baselines. (a) Normalized functional error. (b) Parameter error: drift law, total and projected onto the dominant excited subspace

S

; totals for CL and online Adam. The projected component contracts while the complementary component changes little; the total error of the unconstrained Adam flow grows.

Figure 11: Ablations of the windowed laws (S1; three seeds,

60\,

s; geometric means). (a) Window length, with

L_{F}T=1

at the measured

L_{F}

. (b) Observer gain under a zero-order and a predictive hold: terminal error (solid), post-transient supremum of

\left\|\tilde{f}\right\|

(dashed), and the residual formula

d_{f}/\sqrt{2k_{f}k_{o}}

evaluated with the sampled

d_{f}

(dotted). (c) Learning gain with least-squares and constant gain matrices. (d) Velocity noise for the windowed laws, CL, and online Adam.

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