Neural Reachable Tubes

• Neural reachable tubes are learned representations of reachable sets that retain Hamilton–Jacobi safety semantics while replacing grid-based solvers with neural function approximators.
• They employ self-supervised neural PDE solvers and operator learning techniques to efficiently infer backward reachable tubes in high-dimensional and parameterized systems.
• Recent advances integrate probabilistic certification and parameterized approaches to verify safe sets and optimize closed-loop control in complex, vision-based scenarios.

Neural reachable tubes are learned representations of reachable sets and reachable tubes that preserve the Hamilton–Jacobi (HJ) reachability semantics of safety, adversarial disturbance, and target-set geometry while replacing or augmenting classical grid-based PDE solvers with neural function approximators. In the dominant formulation, a backward reachable tube (BRT) remains the zero sublevel set of a value function, but the value function is represented by a neural network or by a learned operator acting on functions rather than by a dense state-space grid. The resulting literature spans self-supervised neural PDE solvers for high-dimensional HJ problems, exact boundary-condition parameterizations, operator-learning approaches for fast BRT inference across obstacle families and hyperparameters, probabilistic verification methods for learned tubes, and parameterized neural reachable tubes for stress-testing closed-loop vision-based controllers (Bansal et al., 2020, Singh et al., 2024, Li et al., 28 Apr 2025, Lin et al., 2023, Chakraborty et al., 2024).

1. Reachability semantics and value-function formulation

The mathematical core of neural reachable tubes is inherited from HJ reachability. For a controlled system with disturbance,

$\dot{x} = f(x,u,d),$

with target or unsafe set represented as

$L=\{x:l(x)\le 0\},$

the BRT is the set of states from which the disturbance can force the system into $L$ within a finite horizon despite the controller’s best action. A standard form used in the literature is

$$\mathcal R(t)=\{x:\forall u(\cdot),\exists d(\cdot),\exists \tau\in[t,T],\ \zeta(x,u,d,t,\tau)\in\mathcal L\},$$

and the corresponding value function is

$$V(x,t)=\inf_{d}\sup_{u}\min_{\tau\in[t,T]} l(\zeta(x,u,d,t,\tau)), \qquad V(x,T)=l(x).$$

The BRT is then the zero sublevel set

$\mathcal R(t)=\{x:V(x,t)\le 0\}.$

Dynamic programming yields the Hamilton–Jacobi–Isaacs variational inequality

$$\min\left\{ D_t V(x,t)+H(x,t),\ l(x)-V(x,t)\right\}=0,$$

with Hamiltonian

$$H(x,t)=\max_{u}\min_{d}\ \nabla V(x,t)\cdot f(x,u,d).$$

This structure is central because neural reachable tubes do not alter the underlying game-theoretic safety semantics; they alter how the value function is represented, learned, or evaluated. In DeepReach, the same HJ formulation is retained for BRTs and for backward reach-avoid tubes (BRATs), including state constraints and bounded disturbances (Bansal et al., 2020).

A closely related safety-control consequence also survives neuralization. Once a value function is available, the safety controller is obtained from its gradient: $$u^*(x,t)=\arg\max_u\min_d \langle \nabla V(x,t), f(x,u,d)\rangle.$$ Accordingly, in the HJ tradition, a neural reachable tube is not merely a classifier over safe and unsafe states; it is a geometric object induced by a learned value function with control-theoretic meaning (Bansal et al., 2020).

2. Neural PDE solvers for high-dimensional reachable tubes

DeepReach established the canonical neural-reachability paradigm by replacing the explicit grid solve with a neural PDE surrogate $V_\theta(x,t)$ trained directly on the HJI variational inequality rather than on labels from a separate solver. Its loss has a terminal-condition term and a PDE-residual term,

$L=\{x:l(x)\le 0\},$0

where

$L=\{x:l(x)\le 0\},$1

and

$L=\{x:l(x)\le 0\},$2

This is a self-supervised scheme: the PDE residual and terminal condition provide supervision, and the model is trained without explicit solution labels (Bansal et al., 2020).

A distinctive architectural choice in DeepReach is the use of sinusoidal activations. The motivation is structural rather than heuristic: the HJ controller and PDE residual both depend on derivatives of $L=\{x:l(x)\le 0\},$3, and piecewise-linear activations such as ReLU may produce derivative structure that is poorly matched to the fine, non-smooth geometry of reachability value functions. The sinusoidal network is intended to represent both the value surface and its gradients accurately (Bansal et al., 2020).

DeepReach’s empirical significance lies in dimension scaling. The paper reports a BRT volume error of about $L=\{x:l(x)\le 0\},$4 on the Air3D benchmark, and then extends the same approach to a 6D two-vehicle joint-state problem, a 9D three-vehicle collision-avoidance game, and a 10D narrow-passage problem with reach-avoid structure. The stated claim is not that the curse of dimensionality is theoretically removed, but that the computational and memory burden grows with the complexity of the reachable-tube surface rather than with the number of cells in a full tensor grid. This is the core sense in which neural reachable tubes are presented as a scalable alternative to classical level-set solvers for high-dimensional nonlinear systems (Bansal et al., 2020).

The higher-dimensional case studies also sharpen the operational interpretation. In the 9D three-vehicle example, the learned tube captures unsafe states arising from genuinely joint interactions that pairwise approximations miss. In the 10D narrow-passage example, the BRAT produces a safety controller that coordinates evasive maneuvers subject to lane and collision constraints. These examples reinforce that neural reachable tubes are intended to preserve the original HJ semantics of worst-case disturbance, state constraints, and safe control synthesis rather than to replace them with a looser predictive notion (Bansal et al., 2020).

3. Architectural variants: exact boundary imposition and operator learning

Two later developments address complementary bottlenecks in neural reachable-tube computation: boundary-condition fidelity and reusable inference across problem families. Representative formulations are summarized below (Bansal et al., 2020, Singh et al., 2024, Li et al., 28 Apr 2025, Chakraborty et al., 2024).

Formulation	Learned object	Distinctive property
DeepReach	$L=\{x:l(x)\le 0\},$5	Self-supervised HJI VI solver with sinusoidal network
ExactBC	$L=\{x:l(x)\le 0\},$6	Exact terminal-condition imposition by construction
HJRNO	Operator $L=\{x:l(x)\le 0\},$7	FNO-based BRT inference across obstacles and hyperparameters
Vision-based NRTs	$L=\{x:l(x)\le 0\},$8	Parameterized tube over state and environment variables

ExactBC targets a specific weakness of vanilla DeepReach: the terminal safety condition $L=\{x:l(x)\le 0\},$9 is imposed only softly through a weighted boundary loss. ExactBC restructures the value function as

$L$0

which guarantees

$L$1

exactly. The boundary loss is then removed, and training uses only the PDE residual. The paper argues that this eliminates the need to balance loss terms and stabilizes optimization by anchoring the backward-in-time solution to the correct terminal surface. It also introduces a pretraining loss

$L$2

to reduce sensitivity near terminal time (Singh et al., 2024).

The reported gains are substantial on challenging problems. On 6D rocket landing, vanilla DeepReach failed to certify any recovered safe volume, DiffModel recovered about $L$3 out of a learned volume of $L$4, and ExactBC with or without pretraining recovered about $L$5 on average. On 9D three-vehicle collision avoidance, the recovered safe volumes were $L$6 for vanilla DeepReach, $L$7 for DiffModel, $L$8 for ExactBC without pretraining, and $L$9 for ExactBC with pretraining, with the latter also reducing variance across random seeds (Singh et al., 2024).

HJRNO addresses a different problem: repeated solution of BRTs across varying obstacle geometries, dynamics, and hyperparameters. Rather than learning one value function, it learns an operator mapping between value functions: $$\mathcal R(t)=\{x:\forall u(\cdot),\exists d(\cdot),\exists \tau\in[t,T],\ \zeta(x,u,d,t,\tau)\in\mathcal L\},$$0 with

$$\mathcal R(t)=\{x:\forall u(\cdot),\exists d(\cdot),\exists \tau\in[t,T],\ \zeta(x,u,d,t,\tau)\in\mathcal L\},$$1

and, when hyperparameters $$\mathcal R(t)=\{x:\forall u(\cdot),\exists d(\cdot),\exists \tau\in[t,T],\ \zeta(x,u,d,t,\tau)\in\mathcal L\},$$2 are present,

$$\mathcal R(t)=\{x:\forall u(\cdot),\exists d(\cdot),\exists \tau\in[t,T],\ \zeta(x,u,d,t,\tau)\in\mathcal L\},$$3

The implementation uses the Fourier Neural Operator (FNO),

$$\mathcal R(t)=\{x:\forall u(\cdot),\exists d(\cdot),\exists \tau\in[t,T],\ \zeta(x,u,d,t,\tau)\in\mathcal L\},$$4

with Fourier-layer operators

$$\mathcal R(t)=\{x:\forall u(\cdot),\exists d(\cdot),\exists \tau\in[t,T],\ \zeta(x,u,d,t,\tau)\in\mathcal L\},$$5

The stated advantage is that operator learning generalizes across obstacle shapes, system hyperparameters, and spatial discretizations instead of relearning each instance independently (Li et al., 28 Apr 2025).

Its main benchmark uses a 4D dynamic Dubins car with state $$\mathcal R(t)=\{x:\forall u(\cdot),\exists d(\cdot),\exists \tau\in[t,T],\ \zeta(x,u,d,t,\tau)\in\mathcal L\},$$6 and dynamics

$$\mathcal R(t)=\{x:\forall u(\cdot),\exists d(\cdot),\exists \tau\in[t,T],\ \zeta(x,u,d,t,\tau)\in\mathcal L\},$$7

Training on $$\mathcal R(t)=\{x:\forall u(\cdot),\exists d(\cdot),\exists \tau\in[t,T],\ \zeta(x,u,d,t,\tau)\in\mathcal L\},$$8 random obstacle shapes and testing on $$\mathcal R(t)=\{x:\forall u(\cdot),\exists d(\cdot),\exists \tau\in[t,T],\ \zeta(x,u,d,t,\tau)\in\mathcal L\},$$9 cases yields a reported average error of $$V(x,t)=\inf_{d}\sup_{u}\min_{\tau\in[t,T]} l(\zeta(x,u,d,t,\tau)), \qquad V(x,T)=l(x).$$0. For hyperparameter generalization, the model is trained on a sparse $$V(x,t)=\inf_{d}\sup_{u}\min_{\tau\in[t,T]} l(\zeta(x,u,d,t,\tau)), \qquad V(x,T)=l(x).$$1 grid and tested on unseen interpolation points. Inference is reported to take about $$V(x,t)=\inf_{d}\sup_{u}\min_{\tau\in[t,T]} l(\zeta(x,u,d,t,\tau)), \qquad V(x,T)=l(x).$$2 seconds, contrasted with classical HJ solves that may take seconds to hours per instance (Li et al., 28 Apr 2025).

4. Verification and probabilistic certification of learned tubes

Neural reachable tubes introduce an unavoidable verification problem: the learned value function is approximate, and even rare outlier errors may invalidate a nominal safe set. The verification framework developed in 2023 formalizes this issue for neural tubes such as those produced by DeepReach and proposes two probabilistic certification methods: robust scenario optimization and split conformal prediction (Lin et al., 2023).

The objective is to recover a set $$V(x,t)=\inf_{d}\sup_{u}\min_{\tau\in[t,T]} l(\zeta(x,u,d,t,\tau)), \qquad V(x,T)=l(x).$$3 for which the true BRT membership error is controlled. For avoidance problems, the target guarantee is

$$V(x,t)=\inf_{d}\sup_{u}\min_{\tau\in[t,T]} l(\zeta(x,u,d,t,\tau)), \qquad V(x,T)=l(x).$$4

The scenario-based procedure samples $$V(x,t)=\inf_{d}\sup_{u}\min_{\tau\in[t,T]} l(\zeta(x,u,d,t,\tau)), \qquad V(x,T)=l(x).$$5 i.i.d. states from $$V(x,t)=\inf_{d}\sup_{u}\min_{\tau\in[t,T]} l(\zeta(x,u,d,t,\tau)), \qquad V(x,T)=l(x).$$6, evaluates empirical violations, and certifies $$V(x,t)=\inf_{d}\sup_{u}\min_{\tau\in[t,T]} l(\zeta(x,u,d,t,\tau)), \qquad V(x,T)=l(x).$$7 when the binomial-tail inequality

$$V(x,t)=\inf_{d}\sup_{u}\min_{\tau\in[t,T]} l(\zeta(x,u,d,t,\tau)), \qquad V(x,T)=l(x).$$8

holds. Then, with probability at least $$V(x,t)=\inf_{d}\sup_{u}\min_{\tau\in[t,T]} l(\zeta(x,u,d,t,\tau)), \qquad V(x,T)=l(x).$$9,

$\mathcal R(t)=\{x:V(x,t)\le 0\}.$0

Here, $\mathcal R(t)=\{x:V(x,t)\le 0\}.$1 is the risk bound and $\mathcal R(t)=\{x:V(x,t)\le 0\}.$2 is the confidence failure probability (Lin et al., 2023).

The same paper proves that split conformal prediction yields the same usable safety certificate in this setting and, more strongly, that split conformal prediction reduces to a robust scenario-optimization problem. The coverage of the safe portion of $\mathcal R(t)=\{x:V(x,t)\le 0\}.$3 is shown to follow

$\mathcal R(t)=\{x:V(x,t)\le 0\}.$4

with mean

$\mathcal R(t)=\{x:V(x,t)\le 0\}.$5

This establishes a direct bridge between conformal uncertainty quantification and scenario optimization in the verification of neural reachable tubes (Lin et al., 2023).

Because outliers can still shrink the certified set severely, the paper also proposes an outlier-adjusted retraining loop. It treats the true induced cost as supervision and minimizes a weighted MSE

$\mathcal R(t)=\{x:V(x,t)\le 0\}.$6

with

$\mathcal R(t)=\{x:V(x,t)\le 0\}.$7

and $\mathcal R(t)=\{x:V(x,t)\le 0\}.$8, thereby downweighting conservative errors and emphasizing optimistic errors that generate unsafe outliers (Lin et al., 2023).

The experiments use stringent settings, repeatedly taking $\mathcal R(t)=\{x:V(x,t)\le 0\}.$9 and target safety level $$\min\left\{ D_t V(x,t)+H(x,t),\ l(x)-V(x,t)\right\}=0,$$0, described as $$\min\left\{ D_t V(x,t)+H(x,t),\ l(x)-V(x,t)\right\}=0,$$1 safety. On 9D multi-vehicle collision avoidance, the verified safe volume increases from $$\min\left\{ D_t V(x,t)+H(x,t),\ l(x)-V(x,t)\right\}=0,$$2 to $$\min\left\{ D_t V(x,t)+H(x,t),\ l(x)-V(x,t)\right\}=0,$$3, a $$\min\left\{ D_t V(x,t)+H(x,t),\ l(x)-V(x,t)\right\}=0,$$4 improvement. On a 6D rocket-landing problem, it increases from $$\min\left\{ D_t V(x,t)+H(x,t),\ l(x)-V(x,t)\right\}=0,$$5 to $$\min\left\{ D_t V(x,t)+H(x,t),\ l(x)-V(x,t)\right\}=0,$$6, a $$\min\left\{ D_t V(x,t)+H(x,t),\ l(x)-V(x,t)\right\}=0,$$7 improvement. On rocket landing with no-go zones, the baseline cannot recover any safe volume at the desired $$\min\left\{ D_t V(x,t)+H(x,t),\ l(x)-V(x,t)\right\}=0,$$8, whereas the outlier-adjusted method recovers safe volume $$\min\left\{ D_t V(x,t)+H(x,t),\ l(x)-V(x,t)\right\}=0,$$9. These results make a central point: learned tubes may be practically useful at high dimension, but their use in safety-critical settings requires verification that explicitly accounts for approximation error (Lin et al., 2023).

5. Parameterized neural reachable tubes for closed-loop failure mining

A distinct line of work uses neural reachable tubes not primarily as online safety sets for a low-dimensional dynamics model, but as parameterized approximations of BRTs for closed-loop systems whose controller depends on high-dimensional visual observations. In that setting, the central difficulty is that failures arise from the interaction of dynamics, perception, and environment, and are therefore not well characterized by controller prediction error alone (Chakraborty et al., 2024).

The closed-loop system is modeled as

$$H(x,t)=\max_{u}\min_{d}\ \nabla V(x,t)\cdot f(x,u,d).$$0

with sensor map

$$H(x,t)=\max_{u}\min_{d}\ \nabla V(x,t)\cdot f(x,u,d).$$1

where $$H(x,t)=\max_{u}\min_{d}\ \nabla V(x,t)\cdot f(x,u,d).$$2 denotes environmental parameters such as time of day or lighting, and the controller acts through

$$H(x,t)=\max_{u}\min_{d}\ \nabla V(x,t)\cdot f(x,u,d).$$3

The failure set is $$H(x,t)=\max_{u}\min_{d}\ \nabla V(x,t)\cdot f(x,u,d).$$4, the value function is learned as $$H(x,t)=\max_{u}\min_{d}\ \nabla V(x,t)\cdot f(x,u,d).$$5, and the neural reachable tube is defined as

$$H(x,t)=\max_{u}\min_{d}\ \nabla V(x,t)\cdot f(x,u,d).$$6

The training mechanism again follows DeepReach-style self-supervision via an HJB-VI loss with Hamiltonian

$$H(x,t)=\max_{u}\min_{d}\ \nabla V(x,t)\cdot f(x,u,d).$$7

The conceptual shift is that the tube is now parameterized jointly by state and environment variables and represents inevitable closed-loop failure under a vision-based controller (Chakraborty et al., 2024).

The main case study is autonomous aircraft taxiing. The dynamics are

$$H(x,t)=\max_{u}\min_{d}\ \nabla V(x,t)\cdot f(x,u,d).$$8

with constant $$H(x,t)=\max_{u}\min_{d}\ \nabla V(x,t)\cdot f(x,u,d).$$9 m/s, and the controller is

$$u^*(x,t)=\arg\max_u\min_d \langle \nabla V(x,t), f(x,u,d)\rangle.$$0

The unsafe set is runway departure,

$$u^*(x,t)=\arg\max_u\min_d \langle \nabla V(x,t), f(x,u,d)\rangle.$$1

with $$u^*(x,t)=\arg\max_u\min_d \langle \nabla V(x,t), f(x,u,d)\rangle.$$2 m for KMWH. Environment parameters include time of day $$u^*(x,t)=\arg\max_u\min_d \langle \nabla V(x,t), f(x,u,d)\rangle.$$3 and cloud conditions $$u^*(x,t)=\arg\max_u\min_d \langle \nabla V(x,t), f(x,u,d)\rangle.$$4 (Chakraborty et al., 2024).

Once learned, the NRT is used for failure mining by rendering images from states inside the tube,

$$u^*(x,t)=\arg\max_u\min_d \langle \nabla V(x,t), f(x,u,d)\rangle.$$5

These images constitute failure-inducing observations at the level of the full closed loop, not merely at the perception-module level. The paper uses them in two ways. The online method trains a failure detector $$u^*(x,t)=\arg\max_u\min_d \langle \nabla V(x,t), f(x,u,d)\rangle.$$6 and switches to a fallback controller,

$$u^*(x,t)=\arg\max_u\min_d \langle \nabla V(x,t), f(x,u,d)\rangle.$$7

with conformal calibration targeting about $$u^*(x,t)=\arg\max_u\min_d \langle \nabla V(x,t), f(x,u,d)\rangle.$$8 recall via $$u^*(x,t)=\arg\max_u\min_d \langle \nabla V(x,t), f(x,u,d)\rangle.$$9. The offline method augments the original training data with mined failures and retrains the controller incrementally (Chakraborty et al., 2024).

The reported effect is a reduction in NRT volume from $V_\theta(x,t)$0 to $V_\theta(x,t)$1, described as about a $V_\theta(x,t)$2 reduction in closed-loop failures, while running online at about $V_\theta(x,t)$3 Hz. Offline retraining decreases NRT volume by about $V_\theta(x,t)$4 and decreases the DNN’s prediction error by about $V_\theta(x,t)$5 over sampled trajectories on a test runway. The paper further argues that prediction-error-based heuristics, ensembling, and SCOD are misaligned with actual closed-loop failure because some large perception errors are corrected by downstream control, while some small errors lead to failure (Chakraborty et al., 2024).

6. Classical context, misconceptions, and current limitations

Neural reachable tubes emerged against a classical reachability background in which tubes are computed as sublevel sets of viscosity solutions of HJ equations or over-approximated by set-propagation methods. In a differential-game formulation, reachable tubes can be recovered as sublevel sets of a value function with running cost, and the cost-limited reachable tube (CRT) generalizes time-limited reachability by replacing the unit running cost with a general performance index. The improved level-set method shows that multiple horizons or admissible costs can be extracted from sublevel sets of a single slice $V_\theta(x,t)$6, reducing storage by approximately a factor of $V_\theta(x,t)$7 when $V_\theta(x,t)$8 tubes would otherwise be stored separately. The paper is explicit, however, that the PDE must still be solved on a grid, so the curse of dimensionality remains (Liao et al., 2021).

For linear continuous-time, time-varying control systems with compact convex uncertainty in initial states and inputs, another classical method over-approximates reachable tubes over compact time intervals using numerical approximations of transition matrices. It is described as convergent of first order and includes a zonotopic variant that uses only linear algebraic operations and yields zonotopic over-approximations (Serry et al., 2021). These classical results clarify what neural reachable tubes are not: they are not exact viscosity-solution grids, and they are not merely symbolic over-approximations. They are learned surrogates for the value function or for the operator that maps problem data to value functions.

A common misconception is that “neural reachable tube” denotes an arbitrary learned safe-set predictor detached from reachability theory. In the main HJ line of work, that is inaccurate. DeepReach, ExactBC, and HJRNO retain the value-function semantics of HJ reachability and differ primarily in solver, parameterization, or operator-learning strategy (Bansal et al., 2020, Singh et al., 2024, Li et al., 28 Apr 2025). Another misconception is that fast neural inference automatically implies certifiable safety. The verification literature argues the opposite: outlier errors are inevitable in learned tubes, and direct probabilistic certification is required to recover usable safe sets with quantified risk and confidence (Lin et al., 2023).

The present limitations are correspondingly structural. DeepReach does not theoretically remove the curse of dimensionality, even if it makes 9D and 10D examples feasible in practice (Bansal et al., 2020). ExactBC identifies soft imposition of terminal safety conditions as a source of inaccurate value functions and inflated tubes, particularly in complex tasks (Singh et al., 2024). HJRNO notes that hyperparameters are encoded by broadcasting scalar values across the input grid, which increases data size and may not scale well with hyperparameter dimensionality (Li et al., 28 Apr 2025). Verification methods provide probabilistic guarantees rather than exact worst-case certificates for the learned tube itself (Lin et al., 2023). In vision-based settings, parameterized NRTs depend on simulator-generated closed-loop data and may require environment-specific models, as indicated by the instability of a single unified TaxiNet model across all runways (Chakraborty et al., 2024).

Taken together, these results suggest a stable encyclopedic characterization of the field. Neural reachable tubes are best understood as HJ-reachability objects whose computational representation has shifted from tensor grids and explicit set propagation to neural function approximation, neural operator learning, and downstream probabilistic certification. Their central technical problem is not defining reachability anew, but preserving reachability semantics under approximation while extending tractability to high-dimensional, parameterized, and perception-in-the-loop systems.