Neural Barriers: An Online Certifiable Learning-enhanced Adaptive High Order Safety Critical Control

Lishuo Pan; Mattia Catellani; An Cao; Lorenzo Sabattini; Nora Ayanian

cs.ROOct 4, 2026

Neural Barriers: An Online Certifiable Learning-enhanced Adaptive High Order Safety Critical Control

Authors: Lishuo Pan, Mattia Catellani, An Cao, Lorenzo Sabattini, Nora Ayanian

Organizations: Department of Computer Science, Brown University, Providence, RI 02912 USA · Department of Sciences and Methods for Engineering, University of Modena and Reggio Emilia, 41121 Modena, Italy · Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125 USA

Abstract

Control barrier functions are an effective model-based tool to formally certify the safety of a system. However, transferring their theoretical guarantees to real-world robotics systems requires high model fidelity. For example, payloads or wind disturbances can cause significant model perturbations to an aerial vehicle, leading to safety compromises. In this work, we propose a certifiable online learning-enhanced robust adaptive control barrier function, which adapts to disturbances using a Neural ODE and quantifies its adaptation uncertainty with conformal prediction. Our approach guarantees safety at all time under unknown time-varying model disturbances. It adopts a conservative strategy when the adaptation uncertainty is high; and efficiently adapts to reduce controller conservativeness as it receives more data. Our approach provides a provable safety guarantee with a probability bound under suitable Lipschitz smoothness assumptions on the underlying model and trajectory. These results demonstrate the potential of our method as a practical safety controller for robotics system operating under model perturbations.

Figures & tables

Fig. 1 : A 38 $\mathrm{g}$ nano quadrotor tracks a figure-eight trajectory while keeping a safe distance from two obstacles, against an $7.6$\mathrm{m}\mathrm{/}\mathrm{s}$$ wind. Our Neural Barrier provides a probabilistic safety guarantee while reducing the controller conservativeness online. — Fig. 1 : A 38 $\mathrm{g}$ nano quadrotor tracks a figure-eight trajectory while keeping a safe distance from two obstacles, against an $7.6$ \mathrm{m}\mathrm{/}\mathrm{s}$$ wind. Our Neural Barrier provides a probabilistic safety guarantee while reducing the controller conservativeness online.

Fig. 2 : System overview of our Neural Barrier controller. The safety control solves the NODE-HO-RaCBF QP (at $100$\mathrm{H}\mathrm{z}$$ ), using the most recently trained residual model (updated at $2$\mathrm{H}\mathrm{z}$$ ) and conformal prediction bounds (updated at $10$\mathrm{H}\mathrm{z}$$ ), to certify the desired control. — Fig. 2 : System overview of our Neural Barrier controller. The safety control solves the NODE-HO-RaCBF QP (at $100$ \mathrm{H}\mathrm{z} $), using the most recently trained residual model (updated at $2$\mathrm{H}\mathrm{z}$ ) and conformal prediction bounds (updated at $10$ \mathrm{H}\mathrm{z}$$ ), to certify the desired control.

Fig. 3 : Once the $k$ -th model is trained, $\mathcal{Q}_{\mathrm{calib}}$ is cleared and the $k$ -th CP starts to quantify its estimation quality. The data collected during its training and prior data in the queue are used for the training of the $(k\!+\!1)$ -th model.

Panel of Figure 4 — Fig. 4 : Typical trajectories of the baseline and our controllers under different model perturbations. single-obstacle “Attractive” residual (top left). single-obstacle “Time-varying” residual (top right) and multi-obstacle “Time-varying” residual (bottom). The Gray sphere is the obstacle. The blue dot represents the initial state, and the red dot represents the current reference. The red cross indicates the collision. Once the robot collided, the trajectory is depicted in a dashed line.

Fig. 5 : Trajectories of offline Neural Barrier controller with confidence interval computed from Thm. 2 . The model perturbations are “Time-varying” with different magnitude

k

Panel of Figure 6 — Fig. 6 : Trajectories of the HO-RaCBF controller with different hyperparameters. The blue dot represents the initial state, and the red dot represents the current reference. The red cross indicates the collision location. Once the robot collided, the trajectory is depicted in a dashed line.

Residual Type	Attractive					Repulsive					Time-varying
Method	$h_{\mathrm{min}}$	$h_{\mathrm{neg}}$	Avg.	Avg.	CP	$h_{\mathrm{min}}$	$h_{\mathrm{neg}}$	Avg.	Avg.	CP	$h_{\mathrm{min}}$	$h_{\mathrm{neg}}$	Avg.	Avg.	CP
			Dist.	sDist.	Cover.			Dist.	sDist.	Cover.			Dist.	sDist.	Cover.
RCBFs No Res.	0.02	0.0%	0.04	0.00	100%	0.02	0.0%	0.04	0.00	100%	0.02	0.0%	0.04	0.00	100%
RCBFs	8.31	0.0%	1.61	1.63	100%	10.19	0.0%	2.96	3.08	100%	5.57	0.0%	1.59	1.61	100%
RaCBFs (Tuned)	0.00	0.0%	0.10	0.00	-	0.0	0.0%	0.85	0.0	-	-0.42	42.5%	0.28	0.05	-
RaCBFs $\Gamma=1$	-0.73	94.1%	0.21	0.12	-	0.87	0.0%	0.35	0.16	-	-1.44	44.3%	0.33	0.19	-

TABLE I : Quantitative results for different residuals in single-obstacle simulation: Comparison between the baseline controllers and our Neural Barrier controllers (The “HO-” prefix is dropped in all methods due to space limit). The statistics are averaged across 10 trials.

Residual Type	Attractive					Repulsive					Time-varying
Method	$h_{\mathrm{min}}$	$h_{\mathrm{neg}}$	Avg.	Avg.	CP	$h_{\mathrm{min}}$	$h_{\mathrm{neg}}$	Avg.	Avg.	CP	$h_{\mathrm{min}}$	$h_{\mathrm{neg}}$	Avg.	Avg.	CP
			Dist.	sDist.	Cover.			Dist.	sDist.	Cover.			Dist.	sDist.	Cover.
RCBFs No Res.	0.04	0.0%	0.34	0.30	100%	0.04	0.0%	0.34	0.30	100%	0.04	0.0%	0.34	0.30	100%
RCBFs	3.03	0.0%	2.44	2.48	100%	2.69	0.0%	3.50	3.65	100%	2.96	0.0%	1.57	1.57	100%
RaCBFs (Tuned)	-0.04	31.9%	0.35	0.32	-	0.06	0.0%	0.34	0.30	-	-0.39	41.4%	0.37	0.34	-
RaCBFs $\Gamma=1$	-1.43	42.5%	0.52	0.48	-	1.57	0.0%	0.87	0.86	-	-0.61	14.9%	0.55	0.50	-

TABLE II : Quantitative results for different residuals in multi-obstacle simulation: Comparison between the baseline controllers and our Neural Barrier controllers (The “HO-” prefix is dropped in all methods due to space limit). The statistics are averaged across 10 trials.

Fig. 7 :

(1\!-\!\alpha)

quantile of nonconformity scores sampled in online Neural Barrier controller. For each trial, a polynomial of degree

8

is fitted to all quantile samples, the mean and

95\%

confidence interval is plotted. The dot on the error bars represents the median, and the ends depict the

25^{\mathrm{th}}

and

75^{\mathrm{th}}

percentiles of the quantiles in a

1.33

\mathrm{s}$$ time window. The statistics are averaged across 10 trials.

Panel of Figure 10 — Fig. 8 : The time series of CP Coverage for our onlione and offline controller in both single- and multi-obstacle instances under different residual types. The statistics are averaged across 10 trials.

Panel of Figure 11 — Fig. 9 : The ground truth residual dynamics of the $x$ -velocity dimension (red), the neural network prediction (gray), and the CP predictive interval (blue) plots. On the left column, the plots are online residual learning results in single-obstacle “Attractive”, “Repulsive”, “Time-varying”, multi-obstacle “Attractive”, “Repulsive”, and “Time-varying”, respectively. On the right column, the plots are the offline residual learning results. Only one trial is depicted to demonstrate the exact coverage of our Neural Barrier controller.

Panel of Figure 12 — Fig. 10 : The top views of $150$ \mathrm{s}$$ trajectories using different safety controllers in the physical experiments. The red arrows indicate the wind’s location and direction (the length of each arrow represents its strength).

Method	$h_{\mathrm{min}}$	$h_{\mathrm{neg}}$	Avg.Dist.	Avg.sDist.
RCBFs - No Wind	0.07	0.00%	0.13	0.13
RCBFs	0.02	0.06%	0.27	0.28
RaCBFs State $Y$	-0.11	53.14%	0.06	0.06
RaCBFs Const. $Y$	-0.14	53.01%	0.06	0.06
Ours Online 500	-0.23	1.43%	0.18	0.17
Ours Online 2400	-0.01	0.38%	0.16	0.16

TABLE III : Quantitative comparison between the baseline controllers and our controllers (The “HO-” prefix is dropped in all methods due to space limit) in the physical experiments. The statistics are averaged across

5

trials.

Panel of Figure 14 — Fig. 11 : Time series of measured and predicted dynamics residual in the velocity dimension, prediction error, and $(1-\alpha)$ quantile used for WCP in the “Online 2400” physical experiments. The $95\%$ confidence intervals are over $5$ trials.

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As the number of autonomous robots continues to grow, safety becomes increasingly important. Control barrier functions (CBFs) provide a theoretically grounded framework for ensuring safety, but existing design methods often face limitations in effectiveness, scalability, or interpretability, and may result in overly conservative safe sets. In this paper, we propose \emph{VertexCBF}, a framework for learning neural CBFs in a scalable, systematic, and explainable way. We approximate the stationary Hamilton--Jacobi value function using a neural network trained via a combination of physics-informed and sparsely supervised learning. By exploiting control-affine dynamics and a convex polytope control set, under which the Hamiltonian is maximized at the control vertices, we efficiently generate supervision points via GPU-parallel vertex-restricted tree search, while a residual architecture guarantees that the learned CBF is never larger than the specified constraint function. We evaluate the method on 15 systems and compare it against relevant baselines, showing that it reliably recovers large safe sets where the baselines are conservative or fail completely. In addition, we perform a hardware experiment in which a mobile robot safely avoids pedestrians using a neural CBF trained with our method.

Abstract

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