Conformal Prediction with Reachability Analysis

Updated 10 February 2026

Conformal prediction with reachability analysis is a framework that combines finite-sample, distribution-free guarantees with set-based reachability to certify system safety.
It integrates surrogate modeling, conformal calibration, and overapproximation techniques to construct validated reachable tubes that account for uncertainty and adversarial shifts.
The approach is applied in safety-critical domains like robotics and autonomous systems, offering robust, probabilistic safety assurances even under high-dimensional and dynamic conditions.

Conformal prediction with reachability analysis is a principled, data-driven framework for quantifying and certifying the safety of complex, stochastic, and often unknown dynamical systems, particularly when traditional model-based reachability is infeasible or distribution shift (sim-to-real gap) is present. The approach combines conformal inference techniques, which provide finite-sample, distribution-free statistical guarantees, with set-propagation or overapproximation methods from reachability analysis. This synergy enables the construction of set-valued “flowpipes” or reachable tubes that encapsulate future system states with a rigorously quantified probability, even in the presence of learned dynamics, high-dimensionality, perception or model errors, or adversarial distributions.

1. Foundational Principles

Reachability analysis seeks to compute, for a dynamical system, the set of all states that can be reached from an initial set under all admissible uncertainties and disturbances within a given time horizon. Conformal prediction, by contrast, provides marginal coverage guarantees for prediction intervals or sets for arbitrary models, using calibration data to ensure with probability at least $1-\alpha$ that future outcomes fall within the constructed sets.

By integrating these, one overapproximates the reachable set of an unknown, uncertain, or black-box process via:

A learned or surrogate model (e.g., neural networks, black-box predictive models),
Statistical calibration of the model’s prediction error using conformal prediction,
Reachable set propagation that inflates the model’s predicted reachable sets by conformal error bounds,
Probabilistic guarantees that hold even under adversarial distribution shifts when robust conformal methods are employed (Hashemi et al., 2024, Hashemi et al., 2023, Ma et al., 3 Feb 2026).

2. Methodological Frameworks

2.1 Surrogate Modeling and Calibration

The workflow typically begins by collecting trajectory data from a simulator or real observations. A deterministic surrogate model $F_\theta$ is learned—usually via supervised regression—to predict future state trajectories from initial conditions (Hashemi et al., 2023, Hashemi et al., 2024). Advanced approaches employ quantile loss functions to target specific quantiles of the prediction error, directly controlling the miscoverage probability of the residuals. As an example, minimizing the pinball (quantile) loss for level $\tau=1-\epsilon$ tunes the predictor for downstream conformal calibration (Hashemi et al., 2024).

2.2 Reachable Set Computation

Once trained, the surrogate is embedded into a reachability analysis using set-propagation tools (e.g., star-sets, zonotopes, Taylor models, or linear lifting as with Koopman approaches (Nath et al., 3 Jan 2026)). The reachable set under the surrogate dynamics, $\hat{\mathcal{R}}(t)$ , is computed for all $t$ . In lifted or transformed latent spaces, propagation can be conducted efficiently and then mapped back to the original state space, with tight overapproximation (Nath et al., 3 Jan 2026).

2.3 Conformal Calibration and Inflation

The key innovation is inflating the surrogate reachable sets by conformal prediction error bounds. This is done by:

Using a separate calibration set, for which the surrogate model’s errors are computed as nonconformity scores (e.g., weighted maximum-absolute error over all trajectory coordinates).
Extracting empirical quantiles of these scores (e.g., $(1-\epsilon)$ -quantile), yielding an inflation radius that guarantees marginal coverage $1-\epsilon$ .
When adversarial or out-of-sample distributional shift is anticipated, robust conformal inference (e.g., $f$ -divergence balls and robust quantile selection) is employed to ensure coverage even under such shifts (Hashemi et al., 2024).
The final validated reachable set is $R_\text{conf}(t) = \hat{\mathcal{R}}(t) \oplus \mathcal{E}$ , where $\mathcal{E}$ is a conformal error set (typically a zonotope or norm-ball determined as above).

An outline of this sequential workflow is provided in (Hashemi et al., 2024, Hashemi et al., 2023, Ma et al., 3 Feb 2026):

Step	Description	Reference
Training	Fit $F_\theta$ on sampled trajectories (possibly quantile loss)	(Hashemi et al., 2024)
Calibration	Compute errors on held-out data; calibrate conformal bounds	(Hashemi et al., 2023)
Reachability	Propagate sets under $F_\theta$ , then inflate by conformal error	(Hashemi et al., 2024)
Certification	Guarantee $P(\forall t: x(t) \in R_\text{conf}(t)) \ge 1-\epsilon$	(Ma et al., 3 Feb 2026)

3. Advanced Topics: Distribution Shift, State-Dependent Bounds, and Outlier Robustness

Research has extended this methodology to handle several advanced scenarios:

Distribution Shift: When deployment-time systems may differ from the simulator (i.e., $\mathcal{P}$ lies in an $f$ -divergence ball around the offline distribution $\mathcal{Q}$ ), robust conformal inference selects quantiles or adjusts miscoverage rates to deliver guarantees holding uniformly for all $P \in \mathcal{P}_{f,\tau}(\mathcal{Q})$ (Hashemi et al., 2024).
State-Dependent Calibration: In vision-in-the-loop systems, perception error can depend strongly on the underlying system state. Partitioning the state space and assigning region-specific conformal bounds (with local significance levels) yields tighter, less conservative reachability certificates. State partitioning can be optimized (e.g., with genetic algorithms) by minimizing a reachability-informed objective, and merging of reachability branches controls computational complexity (Geng et al., 2 Dec 2025).
Outlier Robustness: Both split conformal and scenario optimization approaches admit trade-offs between coverage confidence and empirical resilience to outliers (violation occurrences). Techniques such as discarding the largest $k$ calibration errors or retraining predictors to focus on worst-case outlier reduction can recover larger safe volumes while maintaining finite-sample probabilistic coverage (Lin et al., 2023).
Alternative Uncertainty Quantification: When a model or forecast provides heuristic uncertainty estimates, conformal calibration (e.g., Rolling Risk Control) can be combined with quantile regression to produce on-the-fly, dynamically feasible confidence intervals for downstream reachability and planning (Muthali et al., 2023).

4. Theoretical Guarantees and Statistical Coverage

The conformal reachability pipeline provides strong theoretical guarantees grounded in the finite-sample, distribution-free properties of conformal prediction:

For any user-prescribed miscoverage $\epsilon$ , the conformal tubes or sets satisfy

$\Pr_{x \sim P}[\forall t: x(t) \in R_\text{conf}(t)] \ge 1-\epsilon$

uniformly over all distributions $P$ in an $f$ -divergence ball around the calibration distribution (Hashemi et al., 2024).

Marginal per-step and trajectory-level guarantees hold, with extensions through union bounds or direct trajectory-level conformal quantiles for reducing conservatism (Ma et al., 3 Feb 2026).
The theory covers both marginal (per-step, per-coordinate) and joint (entire trajectory, region) guarantees, and can handle ensemble methods for value-function certification with a Beta-posterior framework (Tabbara et al., 11 Nov 2025).

No distributional or smoothness assumptions about the underlying stochastic process are required beyond i.i.d. sampling for calibration (Hashemi et al., 2023, Hashemi et al., 2024).

5. Applications and Computational Aspects

Conformal prediction with reachability analysis has been applied across diverse domains:

Learning-enabled safety-critical CPS: Black-box or simulation-based cyber-physical systems where physical modeling is impractical or the sim-to-real gap is significant (Hashemi et al., 2024, Hashemi et al., 2023).
Robotics and autonomous systems: Bipedal locomotion under terrain uncertainty, with Gaussian-process terrain modeling and contraction-based tube reachability, leveraging conformal intervals for probabilistic safety (Muenprasitivej et al., 9 Oct 2025).
Multi-agent planning: Real-time collision avoidance in autonomous driving and aviation, where multi-agent forecasts and conformalized reachability tubes guarantee collision risk bounds (Muthali et al., 2023).
High-dimensional control: Rocket landing, multi-vehicle collision avoidance, and complex reinforcement learning tasks, with neural reachable tubes or Koopman-lifted reachability (Lin et al., 2023, Nath et al., 3 Jan 2026).
Perception-in-the-loop verification: Systems with deep-learning-based perception components, where state-dependent or dynamic conformal calibration is critical for reducing conservatism in time-series reachability (Geng et al., 2 Dec 2025).

Computational aspects depend on the choice of surrogate model (neural vs. polynomial), calibration set size (affecting statistical efficiency), and the reachability engine. Recent works emphasize the scalability of split conformal methods (requiring only quantile selection and residual computation), the composability with high-performance reachability solvers, and the feasibility of online deployment (e.g., for real-time control rates, limited branch merging).

6. Connections to Scenario Optimization and Statistical Verification

Recent work has established that split conformal prediction and robust scenario optimization are fundamentally equivalent for the verification of probabilistic reachable tubes (Lin et al., 2023). Both identify a coverage quantile for residuals/outliers, then condition probabilistic guarantees on the observed violation count using binomial (or Beta) tail bounds. This connection unifies much of the literature under a common statistical framework and clarifies confidence/confidence-level trade-offs, as well as the interpretability of ensemble safety filters and the aggregation of per-initial-state safe probabilities (Tabbara et al., 11 Nov 2025).

7. Limitations and Ongoing Challenges

While conformal prediction with reachability analysis is highly general, its effectiveness is shaped by:

Calibration sample complexity, especially for high-dimensional flows (union-bound conservatism),
The quality and representativeness of simulation/calibration data relative to true deployment distributions,
Possible distribution shift not encapsulated by $f$ -divergence or network support,
Computational overhead of set propagation in high dimensions or under complex nonlinearities,
The intricacy of branch management in state-dependent calibration, and the need for scalable partition optimization,
Balancing conservatism, resilience to outliers, and attainable set size in adversarial or worst-case regimes (Geng et al., 2 Dec 2025, Lin et al., 2023).

Continued research aims to further reduce conservatism—especially for long-horizon or high-dimensional applications—via state-aware calibration, adaptive branch merging, and deeper integration of learning-based and symbolic reachability paradigms.

References

"Statistical Reachability Analysis of Stochastic Cyber-Physical Systems under Distribution Shift" (Hashemi et al., 2024)
"Data-Driven Reachability Analysis of Stochastic Dynamical Systems with Conformal Inference" (Hashemi et al., 2023)
"Conformal Reachability for Safe Control in Unknown Environments" (Ma et al., 3 Feb 2026)
"Verification of Neural Reachable Tubes via Scenario Optimization and Conformal Prediction" (Lin et al., 2023)
"Statistically Assuring Safety of Control Systems using Ensembles of Safety Filters and Conformal Prediction" (Tabbara et al., 11 Nov 2025)
"Probabilistically-Safe Bipedal Navigation over Uncertain Terrain via Conformal Prediction and Contraction Analysis" (Muenprasitivej et al., 9 Oct 2025)
"Statistical-Symbolic Verification of Perception-Based Autonomous Systems using State-Dependent Conformal Prediction" (Geng et al., 2 Dec 2025)
"Scalable Data-Driven Reachability Analysis and Control via Koopman Operators with Conformal Coverage Guarantees" (Nath et al., 3 Jan 2026)
"Multi-Agent Reachability Calibration with Conformal Prediction" (Muthali et al., 2023)
"Data-driven Reachability using Christoffel Functions and Conformal Prediction" (Tebjou et al., 2023)