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)
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)
0.251 (4)
1.984 (0)
1.409 (0)
2.015 (0)
0.064 (2)
0.215 (0)
NODE-replay (control)
0.183 (0)
0.141 (3)
0.311 (0)
0.741 (1)
1.002 (2)
2.129 (1)
0.077 (2)
0.088 (2)
NODE-CL (proposed)
0.035 (5)
0.038 (2)
0.381 (1)
0.302 (4)
1.545 (2)
1.279 (2)
0.069 (1)
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 σv=3×10−3 (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.
S1 (teacher, p=25 )
S2 (manipulator, p=86 )
cost (ms)
Method
func. error [range]
θ~(tf)/θ~(0)
func. error [range]
pred. ( 4 s)
S1
S2
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, kf=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, σv=3×10−3 ) 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 3×10−3 (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 α=2 : terminal functional error and the smallest eigenvalue of Γ 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 ( θˉ=∥θtr∥ , ϵθ=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 12
Figure 11: Ablations of the windowed laws (S1; three seeds, 60 s; geometric means). (a) Window length, with LFT=1 at the measured LF . (b) Observer gain under a zero-order and a predictive hold: terminal error (solid), post-transient supremum of f~ (dashed), and the residual formula df/2kfko evaluated with the sampled df (dotted). (c) Learning gain with least-squares and constant gain matrices. (d) Velocity noise for the windowed laws, CL, and online Adam.
Neural ordinary differential equations (Neural ODEs) often fit training trajectories while generalizing poorly to unseen initial conditions and long horizons. We propose MPINeuralODE, which combines a soft physics-informed residual with a Multiple-Initial-Condition (MIC) multiple-shooting curriculum whose ingredients are structurally complementary: the physics term anchors the vector-field magnitude on the support that MIC enlarges. We evaluate along three axes: out-of-sample error, long-horizon stability, and Hamiltonian drift, which together expose whether the learned dynamics recover the underlying vector field. On Lotka-Volterra, MPINeuralODE achieves the lowest out-of-sample and long-horizon MSE among data-driven methods, with a 24% reduction over the baseline Neural ODE, while essentially matching the PINN ablation on Hamiltonian drift.
Lake Yang, Antonio Malpica-Morales, Frank Ioannis Papadakis Wood +1
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, United Kingdom
This paper presents a learning-based approach for all-pairs motion planning, where the initial and goal states are allowed to be arbitrary points in a safe set. We construct smooth goal-conditioned neural ordinary differential equations (neural ODEs) via bi-Lipschitz diffeomorphisms. Theoretical results show that the proposed model can provide guarantees of global exponential stability and safety (safe set forward invariance) regardless of goal location. Moreover, explicit bounds on convergence rate, tracking error, and vector field magnitude are established. Our approach admits a tractable learning implementation using bi-Lipschitz neural networks and can incorporate demonstration data. We illustrate the effectiveness of the proposed method on a 2D corridor navigation task.
Dechuan Liu, Ruigang Wang, Ian R. Manchester
Australian Centre for Robotics (ACFR), and the School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, NSW 2006, Australia
Offline reinforcement learning (RL) can fail spectacularly when bootstrapped temporal-difference (TD) updates amplify their own errors, driving the critic toward extreme and unusable Q-values. A key counterintuitive insight of this work is that collapse is not only a property of the backup rule or network architecture: optimizer dynamics themselves can directly trigger or suppress instability. From a control-theoretic viewpoint, we model offline TD learning as a feedback system and analyze Adam-based critic updates. This yields a necessary and sufficient condition for stability of the induced local update dynamics: within the regime we analyze, these dynamics are stable if and only if the spectral radius of the corresponding update operator is strictly below one. Further analysis suggests that standard Adam updates can inadvertently distort the parameter geometry, motivating explicit orthogonality constraints to prevent TD error amplification. To this end, we propose AdamO, an Adam-based optimizer with a decoupled orthogonality correction regulated by a strict task-alignment budget. We prove that this design theoretically guarantees worst-case task safety and preserves Adam's continuous-time dissipative dynamics. Empirically, AdamO is broadly compatible with diverse offline RL baselines, improving stability and returns across a broad suite of benchmarks.
Nan Qiao, Sheng Yue, Shuning Wang +1
1Tsinghua University, China · 2Central South University, China · 3Sun Yat-sen University, China