Beyond Testing: Rigorously Verifying AI Spacecraft Control

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Author: Denis Avetisyan

New research details a method for mathematically guaranteeing the safety of neural network controllers used in spacecraft guidance, moving beyond traditional simulation-based verification.

This work leverages Taylor polynomial expansions, interval arithmetic, and domain splitting to provide rigorous safety bounds for neural network controllers without relying on extensive sampling.

Despite increasing reliance on artificial neural networks for spacecraft guidance, a lack of formal verification hinders their deployment in safety-critical applications. This paper, ‘Sample-Free Safety Assessment of Neural Network Controllers via Taylor Methods’, introduces a novel methodology for rigorously assessing the safety of these controllers by leveraging high-order Taylor polynomial expansions, interval arithmetic, and automatic domain splitting to bound potential closed-loop system outcomes. The approach projects state uncertainties onto an event manifold, enabling robust constraint of possible trajectories without relying on extensive simulations. Could this framework unlock broader adoption of neural network controllers in missions demanding guaranteed safety and reliability?

Navigating Complexity: The Limits of Simplified Control

Historically, engineering control systems have frequently depended on representations of reality that prioritize mathematical simplicity over complete accuracy. These simplified models, while easing the burden of calculation, inherently omit crucial details of the controlled system-details encompassing friction, flexibility, or even minor variations in component properties. Consequently, a controller designed based on this incomplete information may perform adequately under ideal conditions, but exhibit significant inaccuracies – or even outright instability – when confronted with the complexities of a real-world environment. This disconnect arises because these approximations fail to capture the nonlinear behaviors and intricate interactions present in dynamic systems, highlighting the limitations of relying solely on linear approximations for effective control, particularly in critical applications where even small deviations can have substantial consequences.

Real-world systems rarely conform to the neat, linear models often used in control design; instead, they are characterized by inherent uncertainties – unpredictable disturbances, parameter variations, and modeling errors – and nonlinearities that defy simple proportional responses. These factors can drastically degrade controller performance, leading to instability or even failure. Consequently, achieving truly robust and reliable control demands techniques capable of explicitly addressing these complexities. Modern control strategies, such as adaptive control and robust control, prioritize performance even with these uncertainties, often employing mathematical frameworks like H_{\in fty} optimization or Lyapunov stability analysis to guarantee stability and desired behavior across a range of operating conditions and potential disturbances. Ignoring these dynamic intricacies risks creating controllers that function adequately in simulations but falter when confronted with the messy realities of physical systems.

Ensuring the reliable operation of complex control systems demands more than just design; it requires exhaustive verification of performance under a multitude of conditions. Contemporary techniques move beyond traditional trial-and-error methods, employing formal verification approaches-rooted in mathematical logic-to rigorously prove controller correctness. These methods can identify potential failure modes and guarantee stability even in the presence of unforeseen disturbances or system uncertainties. Furthermore, advancements in computational power enable simulations that explore an expansive state space, allowing engineers to proactively assess controller behavior before physical implementation. This proactive analysis minimizes risks, reduces development costs, and ultimately leads to more dependable and robust systems, critical in applications ranging from aerospace engineering to autonomous robotics and even medical devices.

Intelligent Control: Bridging Neural Networks and Formal Rigor

Feedback controllers implemented as deep neural networks offer advantages in adaptability and expressiveness compared to traditional control methods. These networks learn control policies directly from data, enabling operation in complex and uncertain environments where explicit modeling is difficult or impossible. The inherent non-linearity of deep neural networks allows for the representation of highly complex control functions, exceeding the capabilities of linear controllers or those based on simpler function approximations. This expressiveness is particularly beneficial in high-dimensional state spaces and for systems with non-linear dynamics, potentially leading to improved performance and robustness. Furthermore, the learned policies can be readily adapted to changing system parameters or operating conditions through retraining, providing a degree of flexibility not typically found in analytically designed controllers.

Sinusoidal Representation Networks (SIREN) layers are employed within the proposed feedback controllers to enhance both performance and generalization. SIREN layers utilize periodic activation functions, specifically sine functions, which allow the network to represent complex functions with fewer parameters compared to traditional ReLU-based networks. This is achieved by mapping inputs to high-dimensional spaces via the sine activations, enabling the network to approximate a wider range of functions and improve its ability to extrapolate beyond the training data. The periodic nature of the activations also contributes to smoother function approximations, which is beneficial for control applications requiring precise and stable behavior.

The integration of deep neural networks with formal verification methods addresses a key limitation of purely data-driven control: guaranteeing system stability and safety. Specifically, polynomial bounding and interval arithmetic are employed to provide provable bounds on the network’s behavior. Interval arithmetic propagates ranges of possible values through each computation, while polynomial bounding approximates the network’s output with a polynomial function. This allows for the creation of a conservative, yet formally verifiable, representation of the network’s input-output relationship. By analyzing this polynomial bound, conditions for stability – ensuring bounded outputs for bounded inputs – and safety – preventing the system from entering undesirable states – can be established and formally proven, offering assurances beyond those achievable with empirical testing alone. This approach enables the application of neural networks to safety-critical systems where predictable and verifiable behavior is paramount.

Precise Boundaries: Guaranteeing Safety Through Analytical Constraints

Taylor methods approximate the dynamics of a system by constructing a polynomial representation of the state variables as a function of time and initial conditions. This approximation utilizes a Taylor series expansion truncated after a finite number of terms, providing a computationally tractable surrogate model. The order of the Taylor expansion directly impacts the accuracy of the approximation; higher-order expansions generally yield better accuracy but require more computational resources. By bounding the remainder term of the Taylor series, polynomial bounds on the state variables can be derived, establishing a region within which the actual system behavior is guaranteed to remain. These polynomial bounds are crucial for safety verification as they provide a conservative estimate of reachable states, independent of simulation time or step size.

Automatic Domain Splitting addresses limitations in analyzing systems over large operating regions by recursively subdividing the state space into smaller, more manageable regions. This process involves monitoring the Taylor method’s approximation error; when error bounds exceed a predefined threshold, the algorithm splits the current region into sub-regions along dimensions where the error is significant. This iterative refinement continues until the error within each sub-region falls below the specified tolerance, allowing for accurate and verifiable bound computation even for systems exhibiting non-linear behavior across expansive state spaces. The technique effectively trades computational complexity for improved accuracy and scalability in formal verification and control design.

Combining polynomial bounds on state variables, derived through methods like Taylor approximation, with Interval Arithmetic provides a rigorous framework for control system verification. Interval Arithmetic replaces each variable and parameter with an interval representing a range of possible values; all arithmetic operations are then performed on these intervals, yielding a guaranteed range for the result. This approach ensures that any potential state of the system, within the defined intervals, remains within pre-defined safety constraints. Consequently, it delivers verifiable guarantees regarding the controller’s stability and its ability to avoid unsafe states, effectively providing a computational proof of safety without requiring exhaustive simulation or testing across all possible scenarios.

Expanding the Horizon: Implications for Space Control and Beyond

This research demonstrates a versatile methodology for assessing the safety of orbital maneuvers, successfully applied to scenarios ranging from the simplified dynamics of relative orbital motion – modeled by the well-established Clohessy-Wiltshire equations – to the significantly more challenging computations involved in Earth-Mars transfer trajectories. The ability to consistently validate safety across such a spectrum of complexity highlights the robustness of the developed techniques. This adaptability is crucial, as it suggests a pathway towards applying these bounding methods to a wider array of space control problems, including collision avoidance, spacecraft formation flying, and active debris removal, irrespective of the underlying orbital dynamics or mission profile.

Analysis reveals a high degree of predictive capability in assessing the safety of orbital maneuvers, with bounding techniques demonstrably covering 92.2% of the initial state-space when applied to the relative motion described by the Clohessy-Wiltshire equations. This substantial safety coverage indicates a robust ability to anticipate potential hazards and prevent collisions, even with considerable uncertainty in initial conditions. The methodology doesn’t simply identify safe trajectories, but actively defines the boundaries of safe operation, providing a quantifiable measure of confidence in system behavior. This level of assurance is critical for applications requiring guaranteed safety, such as proximity operations, satellite servicing, and the prevention of space debris creation, effectively minimizing risk throughout the mission lifecycle.

Event Maps offer a proactive safety measure by explicitly outlining and continuously monitoring critical occurrences within a space control scenario. This technique moves beyond simply predicting trajectories; it defines specific conditions – such as proximity alerts, potential collisions, or exceeding velocity thresholds – as discrete ‘events’ within the operational domain. By meticulously tracking these events, the system can immediately identify and respond to deviations from safe operating parameters. The resulting map provides a clear visualization of potential hazards, allowing for real-time assessment and intervention. This approach enhances safety assurance because it doesn’t rely solely on probabilistic calculations or interval-based estimations, but rather on the explicit identification and continuous surveillance of key risk factors, ultimately bolstering the reliability of space operations.

A rigorous safety evaluation of complex orbital maneuvers, such as Earth-Mars transfers, demands methods capable of handling inherent uncertainties. This work bolsters safety assessments by integrating complementary techniques – Monte Carlo Simulation and Interval Arithmetic – to provide a robust and verifiable analysis. Utilizing automatic domain splitting, the initial problem space is intelligently divided into 478 subdomains, allowing for a more focused and accurate evaluation of potential risks within each region. This granular approach, combined with the strengths of both simulation and interval-based calculations, successfully constrains velocity tolerance violations to a maximum of 296.25 m/s, demonstrating a high degree of confidence in the predicted trajectory and mitigating potential hazards during mission execution.

The pursuit of verifiable safety, as demonstrated in this research concerning neural network controllers, echoes a fundamental principle of robust system design. The methodology, leveraging Taylor polynomial expansions and interval arithmetic to delineate reachable states, aligns with the notion that structure dictates behavior. By meticulously bounding potential outcomes and identifying unsafe regions, the work prioritizes clarity and predictability. As Albert Einstein once observed, “Everything should be made as simple as possible, but no simpler.” This sentiment perfectly captures the essence of the paper’s approach: a dedication to elegant design achieved through rigorous analysis and a refusal to compromise on the essential guarantees of a safe control system.

Beyond Guarantee: Charting a Course for Robust Control

The pursuit of verifiable guarantees for neural network control, as demonstrated by this work, inevitably reveals the hidden costs of complexity. Each added layer of abstraction, each attempt to capture nuanced behavior, introduces a corresponding increase in the difficulty of rigorous validation. The presented methodology, while a significant step forward, operates within the confines of polynomial approximation – a tool whose efficacy diminishes rapidly with system dimensionality and nonlinearity. The challenge, therefore, isn’t merely to refine the bounding techniques, but to reconsider the fundamental architecture of control itself.

Future work must address the interplay between verification and learning. A static, post-hoc analysis, however thorough, can only certify performance within a defined operating envelope. True robustness demands controllers that learn to be verifiable, incorporating constraints that facilitate formal analysis directly into the training process. This necessitates a shift from treating safety as an afterthought to embedding it as an intrinsic property of the system’s design.

Ultimately, the field will be defined not by the sophistication of its verification tools, but by the elegance of its control strategies. The goal isn’t simply to prove a system safe, but to build a system that is inherently safe, a structure where predictable behavior emerges naturally from a simple, understandable core. Every new dependency is the hidden cost of freedom, and the path to reliable autonomy lies in minimizing that cost.

Original article: https://arxiv.org/pdf/2602.11332.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-02-15 19:36