Conformal Reachability: Analysis & Applications

• Conformal reachability is a family of methods that integrate geometric deformation, statistical calibration, and empirical data conformance to produce robust reachability bounds.
• It encompasses geometric notions via conformal metrics, probabilistic prediction with finite-sample guarantees, and hybrid approaches enforcing reachset conformance.
• Applications range from safe dynamical system verification to robotic control, offering practical methods to calibrate and certify uncertainty in reachability analysis.

Searching arXiv for papers on "conformal reachability" and related formulations. Conformal reachability is a family of concepts at the interface of geometry, verification, reachability analysis, and uncertainty quantification. In current arXiv usage, the term does not denote a single universally standardized construction. Instead, it appears in at least three technically distinct senses: a geometric notion of “conformal reach” for geodesics under conformally deformed metrics; a probabilistic reachability paradigm in which reachable sets are calibrated by conformal prediction to achieve finite-sample coverage guarantees; and a data-conformance notion in which reachable sets of an abstract model are required to enclose observed behaviors. Across these strands, the common motif is that reachability is not treated as a purely nominal object: it is regularized, calibrated, or otherwise constrained by an auxiliary structure—respectively a conformal factor, conformal prediction, or empirical conformance (Taupin, 18 Feb 2026, Ma et al., 3 Feb 2026, Tang et al., 2024).

1. Geometric conformal reach and conformal reachability

In geometric analysis, conformal reachability arises from conformal deformation of length metrics on subsets of Euclidean space. The setting in "Estimation of Conformal Metrics" (Taupin, 18 Feb 2026) is a closed, path-connected subset $M \subset \mathbb{R}^N$ with positive reach, together with a conformal factor

$f : M \to (0,\infty),$

assumed Lipschitz with constant $\kappa$ and bounded below by $\underline f>0$. The conformal distance is the induced length metric

$$D_{M,f}(x,y) = \inf_{\gamma\in\Gamma_M(x,y)}\int_I f(\gamma(t))\|\dot\gamma(t)\|\,dt,$$

with minimizing geodesics in $\Gamma^\star_{M,f}(x,y)$ (Taupin, 18 Feb 2026).

The central geometric quantity is the conformal reach

$$\tau_{M,f} := \inf_{\gamma\in\Gamma^\star_{M,f}}\tau_\gamma,$$

the minimal Euclidean reach over all geodesics of the conformal metric. This admits a metric-distortion characterization analogous to the Boissonnat–Lieutier characterization of ordinary reach: $$\tau_{M,f} = \sup\left\{r>0 : \forall x,y\in M,\ \forall \gamma\in\Gamma^\star_{M,f}(x,y),\ \|x-y\|<2r \Rightarrow |\gamma|\le 2r\arcsin\left(\frac{\|x-y\|}{2r}\right) \right\}.$$ In this sense, conformal reach measures how “thick” or “non-bending” conformal geodesics remain when viewed in Euclidean ambient space (Taupin, 18 Feb 2026).

The main conformal reachability statement is quantitative: if $M$ has positive reach $\tau_M>0$, and $f : M \to (0,\infty),$0 is $f : M \to (0,\infty),$1-Lipschitz with $f : M \to (0,\infty),$2, then every conformal geodesic has positive reach and

$f : M \to (0,\infty),$3

This means that conformal deformation preserves a baseline regularity of geodesics: any two points in $f : M \to (0,\infty),$4 are joined by conformal geodesics whose Euclidean geometry satisfies a uniform reach lower bound controlled by $f : M \to (0,\infty),$5, $f : M \to (0,\infty),$6, and $f : M \to (0,\infty),$7 (Taupin, 18 Feb 2026).

A direct consequence is $f : M \to (0,\infty),$8 regularity of conformal geodesics. For an arc-length parametrized simple curve $f : M \to (0,\infty),$9 with reach $\kappa$0,

$\kappa$1

so conformal geodesics have uniformly controlled angular variation and curvature. The paper explicitly interprets this as the core geometric meaning of conformal reachability: geodesics of $\kappa$2 are not only minimizing, but possess quantitatively controlled curvature and thickness (Taupin, 18 Feb 2026).

This geometric strand is not about control-theoretic reachable sets. It concerns the stability and learnability of conformal geodesic structure. The same lower bound $\kappa$3 controls local approximations of conformal distance by straight-segment weighted lengths and underpins graph-based estimation of $\kappa$4 from point clouds. In particular, for point-cloud Hausdorff distance $\kappa$5, graph estimators of the conformal metric achieve relative error proportional to $\kappa$6, and for i.i.d. sampling from a $\kappa$7-standard measure, expected error of order $\kappa$8 is obtained on ball graphs and also on $\kappa$9-NN graphs under Ahlfors regularity (Taupin, 18 Feb 2026).

2. Probabilistic conformal reachability for dynamical systems

In systems and verification, conformal reachability typically means that reachable sets are calibrated by conformal prediction or closely related PAC-style procedures so that future trajectories are contained with prescribed probability. This usage appears in several recent formulations, including data-driven stochastic flowpipes (Hashemi et al., 2023), Koopman-based reachability (Nath et al., 3 Jan 2026), Transformer-accelerated reachability (Zhang et al., 2 Apr 2026), output reachability from input-output data (Zhang et al., 2 Apr 2026), state-dependent perception-aware verification (Geng et al., 2 Dec 2025), and LTT-calibrated data-driven reachability (Huang et al., 12 Mar 2026).

A common structure is the following. First, one computes or learns a nominal reachable-set surrogate: a graph metric, zonotopic propagation, a Koopman-linearized tube, a Transformer-predicted reachable family, or a neural surrogate of trajectory evolution. Second, one defines a nonconformity score measuring discrepancy between predictions and realized trajectories. Third, one calibrates a threshold from held-out data and inflates the nominal set accordingly. The output is a family of reachable sets with finite-sample statistical coverage guarantees under exchangeability or i.i.d. assumptions (Nath et al., 3 Jan 2026, Zhang et al., 2 Apr 2026, Zhang et al., 2 Apr 2026, Huang et al., 12 Mar 2026).

The terminology “conformal reachable set” is explicit in several papers. In the Koopman-based framework "Scalable Data-Driven Reachability Analysis and Control via Koopman Operators with Conformal Coverage Guarantees" (Nath et al., 3 Jan 2026), the authors define a Koopman reachable set (KRS) as a deterministic overapproximation for the decoded Koopman closed-loop model, then define the conformalized Koopman reachable set (CKRS) by Minkowski addition of an error box

$\underline f>0$0

obtained from conformal calibration of model residuals. The resulting tube

$\underline f>0$1

satisfies

$\underline f>0$2

for a new reference trajectory and perturbed initial condition drawn from the same distribution as calibration data (Nath et al., 3 Jan 2026).

A related but distinct data-driven stochastic formulation in "Data-Driven Reachability Analysis of Stochastic Dynamical Systems with Conformal Inference" (Hashemi et al., 2023) constructs a probabilistic flowpipe in full trajectory space. A deterministic surrogate $\underline f>0$3 predicts the entire $\underline f>0$4-step trajectory from the initial state, neural-network reachability yields a surrogate flowpipe $\underline f>0$5, and componentwise conformal residual bounds $\underline f>0$6 define an inflation zonotope

$\underline f>0$7

The final flowpipe

$\underline f>0$8

is shown to be a $\underline f>0$9-confident flowpipe with

$$D_{M,f}(x,y) = \inf_{\gamma\in\Gamma_M(x,y)}\int_I f(\gamma(t))\|\dot\gamma(t)\|\,dt,$$0

so the entire $$D_{M,f}(x,y) = \inf_{\gamma\in\Gamma_M(x,y)}\int_I f(\gamma(t))\|\dot\gamma(t)\|\,dt,$$1-step true trajectory lies in $$D_{M,f}(x,y) = \inf_{\gamma\in\Gamma_M(x,y)}\int_I f(\gamma(t))\|\dot\gamma(t)\|\,dt,$$2 with probability at least $$D_{M,f}(x,y) = \inf_{\gamma\in\Gamma_M(x,y)}\int_I f(\gamma(t))\|\dot\gamma(t)\|\,dt,$$3 (Hashemi et al., 2023). This guarantee is trajectory-level but conservative because it uses a union bound over all state components and time indices.

A more PAC-oriented version appears in "Conformalized Data-Driven Reachability Analysis with PAC Guarantees" (Huang et al., 12 Mar 2026). There, a learned linear or locally affine predictor generates one-step residuals, a scalar score such as $$D_{M,f}(x,y) = \inf_{\gamma\in\Gamma_M(x,y)}\int_I f(\gamma(t))\|\dot\gamma(t)\|\,dt,$$4 is calibrated at each horizon step via Learn Then Test (LTT), and zonotopic reachable sets are propagated using calibrated error sets

$$D_{M,f}(x,y) = \inf_{\gamma\in\Gamma_M(x,y)}\int_I f(\gamma(t))\|\dot\gamma(t)\|\,dt,$$5

or a normalized anisotropic variant. The main guarantee is two-level: $$D_{M,f}(x,y) = \inf_{\gamma\in\Gamma_M(x,y)}\int_I f(\gamma(t))\|\dot\gamma(t)\|\,dt,$$6 This is not merely marginal coverage of one state at one time; it is a trajectory-level PAC statement over calibration draws (Huang et al., 12 Mar 2026).

The diffusion-based formulation "Data-Driven Reachability Analysis via Diffusion Models with PAC Guarantees" (Huang et al., 31 Mar 2026) is even closer to conformal prediction in spirit. At each discrete time $$D_{M,f}(x,y) = \inf_{\gamma\in\Gamma_M(x,y)}\int_I f(\gamma(t))\|\dot\gamma(t)\|\,dt,$$7, a DDPM learns the state distribution $$D_{M,f}(x,y) = \inf_{\gamma\in\Gamma_M(x,y)}\int_I f(\gamma(t))\|\dot\gamma(t)\|\,dt,$$8, and a diffusion-reconstruction nonconformity score

$$D_{M,f}(x,y) = \inf_{\gamma\in\Gamma_M(x,y)}\int_I f(\gamma(t))\|\dot\gamma(t)\|\,dt,$$9

is calibrated via LTT. The reachable set is the sublevel set

$\Gamma^\star_{M,f}(x,y)$0

and the paper proves

$\Gamma^\star_{M,f}(x,y)$1

This is a direct instance of conformal reachability as score-threshold calibration of time-indexed reachable sets (Huang et al., 31 Mar 2026).

3. Data-driven reachability with conformal coverage guarantees

A major branch of the literature treats conformal reachability as a way to calibrate learned or data-driven reachable-set surrogates when explicit models or noise bounds are unavailable. The resulting guarantees are generally probabilistic rather than worst-case.

The most systematic recent formulations are based on matrix-zonotope propagation for unknown linear systems. In "Transformer-Accelerated Interpolated Data-Driven Reachability Analysis from Noisy Data" (Zhang et al., 2 Apr 2026), the baseline data-driven operator propagates zonotopic outer approximations from a single noisy input-state trajectory. The proposed Interpolated Reachability Analysis (IRA) builds coarse anchor sets and reconstructs intermediate sets in parallel. Deterministic outer-approximation guarantees hold for IRA itself: $\Gamma^\star_{M,f}(x,y)$2 Transformer-Accelerated IRA (TA-IRA) then replaces the fine-phase set-valued propagation with an encoder-decoder Transformer and re-establishes guarantees using split conformal prediction. Two coverage modes are defined: pointwise-in-time and path-wise. In pointwise mode,

$\Gamma^\star_{M,f}(x,y)$3

while path-wise mode gives joint interval coverage over all fine substeps in a coarse interval (Zhang et al., 2 Apr 2026).

A related output-only counterpart appears in "Transformer-Enhanced Data-Driven Output Reachability with Conformal Coverage Guarantees" (Zhang et al., 2 Apr 2026). There, the latent state is eliminated using the Cayley–Hamilton theorem, producing an autoregressive lifted model with parameter uncertainty represented by a matrix zonotope. Formal set-valued propagation yields deterministic output reachable sets $\Gamma^\star_{M,f}(x,y)$4, but these are structurally conservative. A decoder-only Transformer predicts tighter zonotopes, and split conformal prediction calibrates them using the score

$\Gamma^\star_{M,f}(x,y)$5

Per-step and joint trajectory-level conformal output reachable sets are then obtained by axis-aligned inflation (Zhang et al., 2 Apr 2026).

The Koopman-based approach (Nath et al., 3 Jan 2026) differs in that the nominal reachability object is already computationally efficient: closed-loop reachability is performed in a learned lifted linear space and verified back in state space using auto_LiRPA/CROWN. Conformal prediction is applied not to the transition itself, but to the model mismatch between the Koopman rollout and the true closed-loop system. This yields reusable error sets across reference trajectories drawn from a planner distribution, a notable distinction from methods requiring recalibration for each new reference (Nath et al., 3 Jan 2026).

These works share two important structural features. First, the set representation is engineered for efficient propagation—zonotopes, interval hulls, linearized lifted spaces, or compact tokenized set representations. Second, conformal calibration is applied after a learned approximation step, not before. In that sense, conformal reachability acts as a statistical wrapper that converts an efficient but uncertified reachable-set predictor into a finite-sample calibrated one (Zhang et al., 2 Apr 2026, Zhang et al., 2 Apr 2026, Nath et al., 3 Jan 2026).

4. State-dependent and multi-agent conformal reachability

A further development is to make conformal uncertainty itself state-, path-, or agent-dependent. This improves tightness when error varies significantly across operating conditions.

In "Statistical-Symbolic Verification of Perception-Based Autonomous Systems using State-Dependent Conformal Prediction" (Geng et al., 2 Dec 2025), perception error $\Gamma^\star_{M,f}(x,y)$6 is bounded by a state-dependent function $\Gamma^\star_{M,f}(x,y)$7, obtained by partitioning the state space into regions $\Gamma^\star_{M,f}(x,y)$8, assigning each region a conformal radius $\Gamma^\star_{M,f}(x,y)$9, and setting

$$\tau_{M,f} := \inf_{\gamma\in\Gamma^\star_{M,f}}\tau_\gamma,$$0

If region-specific miscoverage levels satisfy $$\tau_{M,f} := \inf_{\gamma\in\Gamma^\star_{M,f}}\tau_\gamma,$$1, then

$$\tau_{M,f} := \inf_{\gamma\in\Gamma^\star_{M,f}}\tau_\gamma,$$2

This state-dependent uncertainty is then injected into symbolic reachability of the closed-loop system. The paper proves that if worst-case reachable sets are computed under the deterministic assumption $$\tau_{M,f} := \inf_{\gamma\in\Gamma^\star_{M,f}}\tau_\gamma,$$3, then those reachable sets contain the true trajectory with probability at least $$\tau_{M,f} := \inf_{\gamma\in\Gamma^\star_{M,f}}\tau_\gamma,$$4 (Geng et al., 2 Dec 2025). This is a particularly clear articulation of conformal reachability: conformal prediction supplies probabilistic disturbance bounds, and reachability converts them into probabilistically sound state tubes.

The multi-agent setting in "Multi-Agent Reachability Calibration with Conformal Prediction" (Muthali et al., 2023) calibrates uncertainty in predicted controls of other agents. Quantile regression produces approximate control intervals, RollingRC adapts them online to target a per-agent miscoverage rate $$\tau_{M,f} := \inf_{\gamma\in\Gamma^\star_{M,f}}\tau_\gamma,$$5, and Hamilton–Jacobi reachability propagates these control intervals into forward reachable tubes. Under a conditional independence assumption across agents, if $$\tau_{M,f} := \inf_{\gamma\in\Gamma^\star_{M,f}}\tau_\gamma,$$6 is the desired overall miscoverage budget and $$\tau_{M,f} := \inf_{\gamma\in\Gamma^\star_{M,f}}\tau_\gamma,$$7 is the number of other agents, the paper sets

$$\tau_{M,f} := \inf_{\gamma\in\Gamma^\star_{M,f}}\tau_\gamma,$$8

This makes the multi-agent risk approximately $$\tau_{M,f} := \inf_{\gamma\in\Gamma^\star_{M,f}}\tau_\gamma,$$9, and the ego planner then avoids the union of the calibrated reachable tubes of the other agents (Muthali et al., 2023).

The state-dependent and multi-agent papers illustrate a broader principle: conformal reachability need not produce a single global inflation radius. It can instead allocate risk across state regions, time indices, or agents, yielding materially tighter reachable sets while preserving formal probabilistic coverage under the corresponding assumptions (Geng et al., 2 Dec 2025, Muthali et al., 2023).

5. Reachset conformance and conformal reachability in hybrid systems

A different but related usage appears in hybrid systems and robotics under the name reachset conformance. In "Efficiently Obtaining Reachset Conformance for the Formal Analysis of Robotic Contact Tasks" (Tang et al., 2024), the issue is not statistical calibration of a learned set predictor, but synthesis of a simple hybrid automaton whose reachable output sets enclose all measured robotic behaviors.

The central conformance condition is

$$\tau_{M,f} = \sup\left\{r>0 : \forall x,y\in M,\ \forall \gamma\in\Gamma^\star_{M,f}(x,y),\ \|x-y\|<2r \Rightarrow |\gamma|\le 2r\arcsin\left(\frac{\|x-y\|}{2r}\right) \right\}.$$0

meaning every measured output of the implementation must lie inside the model’s reachable output set at the corresponding time under the same input (Tang et al., 2024). The hybrid automaton has affine-linear continuous dynamics within each mode, with zonotopic uncertainty injected into flows, outputs, and resets: $$\tau_{M,f} = \sup\left\{r>0 : \forall x,y\in M,\ \forall \gamma\in\Gamma^\star_{M,f}(x,y),\ \|x-y\|<2r \Rightarrow |\gamma|\le 2r\arcsin\left(\frac{\|x-y\|}{2r}\right) \right\}.$$1 Model parameters and zonotopic uncertainty sets are jointly optimized so that all observed trajectories are contained while a norm-based cost of reachable-set size is minimized (Tang et al., 2024).

This is not conformal prediction, and it does not rely on finite-sample exchangeability arguments. Nevertheless, it instantiates a notion of conformal reachability in the broader sense that reachability is calibrated to data: reachable sets are forced to conform to observed trajectories. The significance is practical and formal. Reachset conformance is presented as necessary and sufficient for transferring safety properties from the abstract model to the robotic implementation, so once conformance is achieved, standard forward reachability on the hybrid model yields sound safety statements for the real robot (Tang et al., 2024).

A plausible implication is that reachset conformance and conformal-prediction-based reachability occupy complementary positions on a spectrum. The former yields deterministic, data-fitting outer approximations tied to a chosen hybrid model class; the latter yields probabilistic coverage guarantees with weaker structural assumptions but typically no deterministic containment of all possible behaviors.

6. Guarantees, trade-offs, and open questions

Recent work emphasizes that “conformal reachability” is not a single guarantee type. Several distinct guarantee semantics coexist.

The first is deterministic outer approximation, as in IRA or output-zonotope propagation: $$\tau_{M,f} = \sup\left\{r>0 : \forall x,y\in M,\ \forall \gamma\in\Gamma^\star_{M,f}(x,y),\ \|x-y\|<2r \Rightarrow |\gamma|\le 2r\arcsin\left(\frac{\|x-y\|}{2r}\right) \right\}.$$2 for all admissible disturbances and inputs (Zhang et al., 2 Apr 2026, Zhang et al., 2 Apr 2026). The second is trajectory-level probabilistic coverage of the true system under exchangeability or i.i.d. sampling, as in CKRS, stochastic flowpipes, and CDDR: $$\tau_{M,f} = \sup\left\{r>0 : \forall x,y\in M,\ \forall \gamma\in\Gamma^\star_{M,f}(x,y),\ \|x-y\|<2r \Rightarrow |\gamma|\le 2r\arcsin\left(\frac{\|x-y\|}{2r}\right) \right\}.$$3 or its two-level PAC analogue (Nath et al., 3 Jan 2026, Hashemi et al., 2023, Huang et al., 12 Mar 2026). The third is training-conditional or PAC coverage over random calibration sets, which is emphasized in LTT-based methods and in the comparison between conformal prediction, holdout, and scenario optimization (Dietrich et al., 3 Apr 2026).

The paper "Probably Approximately Correct (PAC) Guarantees for Data-Driven Reachability Analysis: A Theoretical and Empirical Comparison" (Dietrich et al., 3 Apr 2026) is especially clarifying on this point. It shows that an empirical conformal coverage construction and the holdout method yield identical PAC bounds on reachable-set violation probability when parameterized appropriately. It also establishes a structural connection between split conformal reachability with training-conditional coverage and scenario optimization with discarding, including a matched-parameter regime in which the two produce identical reachable sets. However, the paper stresses that the guarantees are interpreted differently and are therefore not generally interchangeable (Dietrich et al., 3 Apr 2026).

Several recurring trade-offs emerge across the literature:

Dimension	Tighter sets	Stronger guarantees
Risk model	Marginal CP or learned surrogate without inflation	Path-wise CP, LTT-PAC, or deterministic outer approximation
Set representation	Learned Transformer, Koopman tubes, diffusion score level sets	Zonotopic envelopes, interval hulls, exact graph propagation
Calibration granularity	State-dependent, agent-dependent, anisotropic normalization	Global thresholds, Bonferroni allocation, worst-case noise sets

Detailed explanations of these trade-offs belong outside the table. State-dependent or normalized scores can dramatically reduce reachable-set volume under heteroskedastic or anisotropic uncertainty, but they often increase model complexity or branching in symbolic propagation (Geng et al., 2 Dec 2025, Huang et al., 12 Mar 2026). Path-wise calibration avoids repeated union bounds and gives stronger trajectory-level semantics, but usually produces larger inflations than pointwise calibration (Zhang et al., 2 Apr 2026, Zhang et al., 2 Apr 2026). Deterministic outer approximations remain stronger than conformal tubes in an adversarial sense, but they rely on more restrictive assumptions such as bounded residuals or known set-valued uncertainty models (Zhang et al., 2 Apr 2026, Zhang et al., 2 Apr 2026).

Open questions are explicit in several sources. In geometric conformal reach, it remains open whether current analysis can close the gap between known lower bounds and achieved rates for induced-metric estimation under positive reach, and whether the bound $$\tau_{M,f} = \sup\left\{r>0 : \forall x,y\in M,\ \forall \gamma\in\Gamma^\star_{M,f}(x,y),\ \|x-y\|<2r \Rightarrow |\gamma|\le 2r\arcsin\left(\frac{\|x-y\|}{2r}\right) \right\}.$$4 can be sharpened (Taupin, 18 Feb 2026). In probabilistic reachability, open issues include reducing Bonferroni conservatism for long horizons (Hashemi et al., 2023), extending guarantees beyond exchangeable settings (Nath et al., 3 Jan 2026), and jointly learning set shape and calibration rather than post-hoc inflating a fixed predictor (Dietrich et al., 3 Apr 2026). In safe control for unknown environments, a central limitation is conservatism of conformal reachable tubes when the learned model is poor or high-dimensional verification becomes expensive (Ma et al., 3 Feb 2026).

Taken together, the current literature suggests that conformal reachability is best understood as an umbrella term for methods that augment reachability with calibrated uncertainty control. In geometric settings, this means explicit lower bounds on the reach of conformal geodesics (Taupin, 18 Feb 2026). In data-driven verification and control, it means reachable sets whose containment guarantees are statistically calibrated, often by conformal prediction or LTT (Huang et al., 12 Mar 2026, Huang et al., 31 Mar 2026, Ma et al., 3 Feb 2026). In hybrid modeling, it can mean reachability calibrated to observed behavior through conformance constraints (Tang et al., 2024). The unifying theme is that reachability is no longer treated as a purely model-derived construct: it is shaped by calibration, deformation, or empirical conformity, with the aim of making safety or geometry both rigorous and learnable.