Defining Operational Conditions for Safety-Critical AI-Based Systems from Data

Abstract: AI has been on the rise in many domains, including numerous safety-critical applications. However, for complex systems found in the real world, or when data already exist, defining the underlying environmental conditions is extremely challenging. This often results in an incomplete description of the environment in which the AI-based system must operate. Nevertheless, this description, called the Operational Design Domain (ODD), is required in many domains for the certification of AI-based systems. Traditionally, the ODD is created in the early stages of the development process, drawing on sophisticated expert knowledge and related standards. This paper presents a novel Safety-by-Design method to a posteriori define the ODD from previously collected data using a multi-dimensional kernel-based representation. This approach is validated through both Monte Carlo methods and a real-world aviation use case for a future safety-critical collision-avoidance system. Moreover, by defining under what conditions two ODDs are equal, the paper shows that the data-driven ODD can equal the original, underlying hidden ODD of the data. Utilizing the novel, Safe-by-Design kernel-based ODD enables future certification of data-driven, safety-critical AI-based systems.

• The paper introduces a novel deterministic, kernel-based method to define operational design domains directly from data, enabling rigorous safety certification.
• The methodology applies local positive-definite kernels with adaptive bandwidths to capture high-dimensional, nonconvex operational manifolds while excluding unsafe regions.
• Real-world validation in aviation demonstrated high statistical reliability (R²>0.97), highlighting its potential for certification in safety-critical AI systems.

Data-Driven Formulation of Operational Conditions for Certifiable Safety-Critical AI Systems

Introduction

The certification and deployment of AI-based systems in safety-critical domains such as aviation and automotive require a rigorous specification of the conditions under which the AI components are claimed to operate safely. This set of conditions is formalized as the Operational Design Domain (ODD). Conventional ODD definition methodologies rely on expert-driven, a priori specification and typically result in conservative or incomplete descriptions that struggle with high-dimensional, real-world parameter dependencies and implicit constraints. The paper “Defining Operational Conditions for Safety-Critical AI-Based Systems from Data” (2601.22118) introduces a deterministic, kernel-based approach for deriving mathematically well-posed, certifiable ODDs directly from data. This methodology supports safety-by-design AI engineering and aligns with regulatory trends demanding provable operational boundaries for certification.

Foundations and State of the Art

Current ODD frameworks formalize operational boundaries through an $n$-dimensional parameter taxonomy and an explicit ontology capturing parameter dependencies. Prior approaches generally deploy convex polytopes, structured tabular representations, or scenario- and scene-based abstractions, but frequently fail to capture nonconvex operational manifolds, high-order constraints, or the inherent uncertainty and sparseness of observed operational distributions. Data-driven techniques have been discussed in relation to scenario generation and runtime monitoring, but determinism, reproducibility, and direct traceability from data to ODD boundaries are lacking.

The proposed framework (2601.22118) bridges this gap through a unique, order-independent, kernel-based ODD construct derived from anchor samples. This addresses the central issues of (i) data-anchored verification of operational coverage, (ii) formal comparison and equivalence between ODD models, and (iii) tractable validation for high-dimensional safety-relevant systems.

Mathematical Formulation and Kernel-Based Construction

Formal ODD Structure

The ODD is formally defined as a tuple %%%%1%%%%, with $X \subseteq \mathbb{R}^n$ denoting the parameter space (taxonomy), and $\mathcal{R}$ a set of Boolean relationships (ontology) constraining admissible operational points. The data set $Y$ is a (potentially stochastic) mapping from admissible parameterizations to observed states. Two ODDs are mathematically equivalent if they generate the same $Y$ under all admissible operating conditions, supporting rigorous structural comparison.

Kernel Affinity Representation

A core innovation is the construction of a global ODD membership affinity function $\alpha: X \to [0,1]$ using an aggregation of local positive-definite kernels, each centered on an anchor point—a representative in-distribution (ID) sample. For an anchor at $x_i$, the local affinity $\alpha_i(x)$ is instantiated, for example, as a diagonal RBF kernel to ensure scalability and computational tractability:

$$\alpha_i(x) = \exp\Big(-\tfrac{1}{2}(x-x_i)^\top \Sigma_i^{-1}(x-x_i)\Big).$$

The global affinity is then

$\alpha(x) = 1 - \prod_i [1 - \alpha_i(x)],$

ensuring smoothness, boundedness, and automatic shrinkage in sparse regions.

Crucially, kernel bandwidth matrices $\Sigma_i$ are parameterized based on local sample density—proportional to the distance to the nearest neighbor—yielding adaptivity while maintaining determinism and ODD boundary conservatism.

Out-of-Distribution Constraints

The method incorporates explicit OOD samples (negative anchors) by enforcing a maximum allowable global affinity at OOD points, tuning dominant kernels to guarantee discrimination and ensure that the reconstructed ODD does not falsely include known-unsafe operational states. This is critical for regulatory alignment and runtime assurance.

Validation and Real-World Application

The proposed approach was validated via Monte Carlo analysis and a high-fidelity aviation use case. In simulation, random anchor points in 2D and up to 10D parameter spaces, coupled with polynomial constraints, demonstrated that the kernel-based ODD closely approximates the true ODD and, importantly, always remains a subset of the convex hull of anchor points, thereby avoiding non-convex voids included by naive convex hull methods. The precision-recall characteristics of the affinity-thresholded ODD closely match those of the ground truth and convex hull boundaries, with $R^2$ values exceeding 0.97 across all tests, establishing statistical reliability.

In a prominent aviation case—the next-generation vertical collision avoidance system (VCAS)—the framework processed 622,110 anchor points across a 5-dimensional operational envelope (relative altitude, ownship/intruder vertical rates, time-to-CPA, advisory history). The resulting ODD affinity function was validated with $10^7$ synthetic operation points, yielding precision-recall curves with $R^2>0.99$ agreement against both engineered groundtruth and convex hull benchmarks. The adaptivity to local sample density and exclusion of non-observable voids was demonstrated.

The following scenario geometry underpins the real-world validation:

(Figure 1)

Figure 1: Geometry of the vertical collision avoidance scenario for VCAS; the ownship (black) maneuvers vertically to avoid a red intruder [Julian2019].

The kernel-based approach can therefore be deployed for real-time monitoring and certification support in AI-based collision avoidance architectures, mitigating deficiencies of expert-driven and convex hull methods.

Discussion and Implications

The kernel-based ODD representation possesses several characteristics advantageous for certifiable safety-critical AI:

• Determinism: The representation is strictly data-driven, order-independent, and not subject to training randomness.
• Conservatism: By tightly wrapping the data manifold, it excludes operational voids that simple convex representations cannot, thus preventing certification on unobserved unsafe states.
• Soft Boundary and Graded Monitoring: The continuous $\alpha(x)$ enables not only safety assertions but graded hazard warnings and safe region degradation monitoring.
• Certifiability: The explicit parameterization and deterministic construction enable integration into certification workflows, linking data coverage with regulatory safety requirements.
• Order- and Data-Independence: Retraining and different data orders do not affect the ODD boundary, which is essential for traceable certification.

A key assertion of the paper is that data-driven ODDs, if sufficiently sampled, may be proven to contain the hidden true operational region of the system. However, coverage is always dependent on dataset representativeness, especially in high-dimensions, and method conservatism ensures no false inclusion of undersampled states.

Future Directions

Potential exploitation directions include:

• Kernel Cross-Dimensionality: While the current formulation leverages diagonal kernels, extension to full covariance models would allow richer encoding of coupled operational dependencies at the expense of interpretability and certifiability.
• Dynamic/Temporal ODDs: Integrating time-series kernels will support time-varying operational envelopes, necessary for dynamic environments.
• Certification Pipeline Integration: Automated threshold selection, traceable ODD evolution with field data, and alignment with regulatory continuous assurance will be key for operational deployment in mission- and safety-critical AI.

Conclusion

This work defines a deterministic, interpretable, data-first method for ODD construction suitable for certifiable AI engineering in safety-critical settings. The kernel-based approach guarantees conservative operational boundary definition, supports high-dimensional and implicit constraints, and facilitates integration into formal certification and runtime monitoring regimes. The methodological foundation and empirical validation suggest broad applicability beyond the demonstrated aviation domain, including automotive, industrial automation, and other areas where certifiable AI behavior is requisite (2601.22118).