Stabilizing Complex Systems with AI

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Author: Denis Avetisyan

A new data-driven approach leverages graph neural networks to guarantee the stability of large-scale interconnected systems, even with unknown underlying dynamics.

Trajectories originating from diverse initial conditions demonstrably converge, accompanied by a decaying Lyapunov value-evidence of $δ$-GAS-even without external intervention, suggesting an inherent stability within the subsystem.
Trajectories originating from diverse initial conditions demonstrably converge, accompanied by a decaying Lyapunov value-evidence of $δ$-GAS-even without external intervention, suggesting an inherent stability within the subsystem.

This work presents a scalable framework for formally verifying incremental stability using Lyapunov functions constructed from graph neural networks.

Guaranteeing the stability of increasingly complex, interconnected systems remains a significant challenge despite decades of control theory research. This paper, ‘Scalable Formal Verification of Incremental Stability in Large-Scale Systems Using Graph Neural Networks’, introduces a data-driven framework leveraging graph neural networks to construct and formally verify Lyapunov functions for ensuring incremental stability, even with unknown system dynamics. By synthesizing local stability estimates and composing them for the entire system, this approach enables scalable and distributed control design. Could this methodology unlock robust control strategies for previously intractable large-scale infrastructures and robotics applications?

Beyond Fixed Points: Embracing Dynamic Stability

Conventional control systems and stability analyses frequently center on the concept of fixed equilibrium points – specific states a system aims to reach and maintain. However, this approach presents limitations when applied to complex, interconnected systems common in fields like power grids, climate modeling, and even biological networks. These systems rarely settle at a single, static point; instead, they exhibit dynamic behavior and constant fluctuations. Relying on fixed points ignores the inherent variability and can lead to instability when faced with disturbances or uncertainties. The assumption of a precisely defined, achievable equilibrium becomes increasingly unrealistic as system complexity grows, necessitating analytical tools that move beyond this traditional paradigm and embrace the reality of dynamic, ever-changing states. This focus on fixed points often overlooks the crucial aspect of relative stability – the ability of trajectories to converge and remain bounded, even without reaching a specific target.

Many complex systems, from ecological networks to financial markets and even the human body, rarely settle at a precise, static equilibrium. Instead, these systems are characterized by ongoing fluctuations and adaptations, where the relationship between different states is paramount. Maintaining stability isn’t about achieving a single target point, but rather ensuring that various internal trajectories – representing different system components or variables – converge and remain predictably related. This ‘relative convergence’ allows the system to absorb disturbances and continue functioning effectively, even when faced with uncertainty or change. For instance, a flock of birds doesn’t maintain a fixed formation; it’s the consistent pattern of movement and the ability of individuals to adjust to each other that defines its collective stability. This principle of maintaining proportional relationships, rather than absolute positions, is increasingly recognized as fundamental to understanding resilience in dynamic, interconnected environments.

Traditional assessments of system stability often prioritize convergence to a singular equilibrium point, a limitation when dealing with the inherent complexities of interconnected networks. However, many natural and engineered systems demonstrate resilience not through absolute stability, but through incremental stability – a capacity for trajectories to remain relatively close, even amidst disturbances or imperfect modeling. This approach acknowledges that complete convergence may be unrealistic or even undesirable, instead focusing on maintaining a bounded divergence between system states. By analyzing the rate at which nearby trajectories converge – or fail to diverge significantly – incremental stability provides a more robust metric for assessing system performance under real-world conditions. This is particularly valuable when facing model uncertainties or external perturbations, as it offers a degree of operational tolerance that point-based stability analyses often lack, ensuring predictable and manageable behavior even when precise equilibrium is unattainable.

Trajectories originating from diverse initial conditions converge, and the associated Lyapunov value consistently decays, demonstrating delta-GAS stability without external disturbances.
Trajectories originating from diverse initial conditions converge, and the associated Lyapunov value consistently decays, demonstrating delta-GAS stability without external disturbances.

Deconstructing Complexity: A Compositional Framework for Stability

The proposed Compositional Framework addresses the challenge of verifying stability in complex systems by decomposing the analysis into smaller, more manageable subsystems. Rather than attempting to directly analyze the interconnected system as a whole, this approach focuses on individually assessing the stability of each subsystem and their local interactions. This decomposition is achieved by treating each subsystem as an independent component with defined inputs and outputs, allowing for separate stability verification using established Lyapunov-based methods. The framework then provides a mechanism to combine these individual subsystem analyses to determine the stability of the overall interconnected system, significantly reducing computational complexity and facilitating scalability for large-scale systems. This approach is predicated on the assumption that local stability of subsystems, combined with appropriate interconnection properties, is sufficient to guarantee global stability.

The Compositional Framework utilizes Local δ-Incremental Stability (δ-ISS) Lyapunov Functions to reduce the computational complexity of stability analysis for interconnected systems. These local functions, $V_i$, are defined solely based on the state variables of subsystem $i$ and its immediate neighbors, eliminating the need to consider the entire system state during verification. This localized approach allows for independent analysis of each subsystem, significantly simplifying the overall process compared to traditional global Lyapunov function methods which require knowledge of the entire system dynamics. The δ-ISS property guarantees that disturbances affecting one subsystem will not lead to instability in neighboring subsystems, provided the disturbances are sufficiently small, and this localized property is directly incorporated into the Lyapunov function structure.

The Compositional Framework establishes a method for constructing a global Lyapunov function, $V(x)$, from a set of local Lyapunov functions, $V_i(x_i)$, each defined for a subsystem $i$ and dependent only on the state of its neighboring subsystems. This construction relies on summing the local Lyapunov functions, ensuring that $V(x) = \sum_i V_i(x_i)$ remains radially unbounded and positive definite for the entire interconnected system. Demonstrating that the time derivative of this global Lyapunov function, $\dot{V}(x)$, is negative definite – or at least negative semi-definite – constitutes a sufficient condition to prove incremental stability of the complete system, without requiring explicit analysis of the full, potentially high-dimensional, system dynamics.

Graph Neural Networks: Scaling Lyapunov Representation

Graph Neural Networks (GNNs) are employed to approximate local Lyapunov functions for interconnected systems due to their capacity to represent relationships between entities. Traditional methods for defining Lyapunov functions require detailed mathematical models of each subsystem and their interactions, which can be computationally expensive or unavailable. GNNs, however, leverage the system’s interconnection structure directly as input; node features represent local state information, and edges defined by the adjacency matrix encode the system’s topology. This allows the GNN to learn a function that estimates the energy or stability of each local subsystem based on its own state and the states of its neighbors, effectively representing a localized Lyapunov function without explicit modeling of the underlying dynamics. The inherent ability of GNNs to propagate information across the graph structure is thus crucial for capturing the interdependencies between subsystems and assessing overall system stability.

The system’s interconnection topology is represented by its Adjacency Matrix, a square matrix where the entry $A_{ij}$ is non-zero if a connection exists from node $i$ to node $j$, and zero otherwise. This matrix is directly integrated into the Graph Neural Network (GNN) architecture as a structural input; specifically, it defines the graph’s edges and facilitates message passing between interconnected nodes. By utilizing the Adjacency Matrix, the GNN can efficiently process information based on the system’s network configuration, enabling scalable representation of local dynamics even in systems with a large number of interconnected components. This approach avoids the computational bottlenecks associated with traditional methods of representing complex interdependencies.

Traditional Lyapunov function construction often necessitates detailed mathematical models of each subsystem within an interconnected system, a process that can be computationally expensive and impractical for complex networks. This GNN-based approach circumvents this requirement by learning Lyapunov functions directly from data representing system behavior. The GNN infers the local dynamics and interdependencies through message passing on the graph, defined by the system’s adjacency matrix, effectively creating an approximation of the Lyapunov function without explicitly solving differential equations or requiring precise state-space representations of individual subsystems. This data-driven method allows for scalability to larger, more complex interconnected systems where analytical modeling is intractable, and facilitates the analysis of system stability even with incomplete or uncertain subsystem models.

Formal Verification: Guaranteeing Stability in the Real World

The development of formally verified systems increasingly relies on a synergy between optimization and verification strategies. This work leverages sampling-based optimization to train Lyapunov functions – mathematical expressions that guarantee system stability – and then employs data-driven verification to provide formal assurances of their correctness. By intelligently sampling the system’s state space, the optimization process efficiently identifies Lyapunov functions that meet desired criteria. Subsequent verification, guided by data collected during system operation, rigorously checks these functions against pre-defined tolerances, effectively establishing formal guarantees of stability and performance. This approach circumvents the limitations of traditional methods, offering a scalable pathway to ensure the reliable operation of complex systems where analytical solutions are intractable.

A crucial component of ensuring the reliability of complex systems lies in formally verifying the smoothness and validity of Lyapunov functions, and this process heavily relies on establishing Lipschitz continuity. This mathematical property guarantees that small changes in the input to the function will result in correspondingly small changes in the output, preventing erratic behavior and ensuring stability analysis remains accurate. Specifically, the analysis utilized Lipschitz constants – values representing the maximum rate of change – of 1.25 for the Temperature Control Network and 1.5 for the Nonlinear 2D System. These constants effectively bound the function’s sensitivity, providing a quantifiable measure of its smoothness and allowing for rigorous, formal verification of system stability; a lower constant indicates a smoother function and potentially faster verification times, while a higher value suggests a more complex, though still verifiable, dynamic.

The efficacy of this formal verification method is demonstrated through its application to both a Temperature Control Network and a more complex Nonlinear 2D System, highlighting its adaptability to varying system complexities. Initial training of the Temperature Network required 45 minutes, while the 2D System necessitated a longer 2-hour period; subsequent fine-tuning to achieve a dataset size of N=1000 added a further 20 and 40 minutes respectively. Critically, verification was successfully completed with tight error tolerances-0.0002 for the Temperature Network and 0.001 for the 2D System-underscoring the precision of the approach and its potential for deployment in safety-critical applications where rigorous guarantees are paramount. These results demonstrate not only the method’s robustness but also its scalability, suggesting its viability for increasingly complex systems.

The pursuit of formal verification, as detailed in this work concerning incremental stability and Graph Neural Networks, feels remarkably cyclical. It’s a grand attempt to impose order on chaos, to predict where systems will break, rather than simply reacting when they inevitably do. As Søren Kierkegaard observed, “Life can only be understood backwards; but it must be lived forwards.” This neatly encapsulates the irony. The framework strives to anticipate instability – a forward-looking endeavor – yet relies on data from past system behavior. It’s elegant, certainly, but one suspects production will, as always, find a novel way to invalidate even the most rigorously verified Lyapunov functions. Everything new is old again, just renamed and still broken.

What’s Next?

The promise of data-driven Lyapunov function construction is, predictably, attracting attention. It will be framed as a solution to problems previously considered intractable. The current framework, however, elegantly sidesteps the issue of certifiable robustness. These Graph Neural Networks will perform admirably on the training distribution, naturally. But the moment production encounters a novel interconnection, or a slightly perturbed dynamic, the carefully constructed Lyapunov function will likely resemble a house of cards. They’ll call it AI and raise funding, of course.

A more pressing concern isn’t scalability, but interpretability. The derived Lyapunov functions are, at present, black boxes. Understanding why a particular function guarantees stability is crucial, not just for safety-critical applications, but for debugging when – not if – things inevitably fail. It started as a simple bash script, everyone swears, but now it’s a distributed system with layers of abstraction. The documentation lied again, predictably.

The real challenge isn’t building fancier neural networks. It’s acknowledging that formal verification, in the face of truly complex systems, is less about absolute certainty and more about gracefully managing uncertainty. Tech debt is just emotional debt with commits, after all. Future work will need to address the gap between mathematical guarantees and empirical performance, lest this framework joins the graveyard of ‘revolutionary’ control techniques.

Original article: https://arxiv.org/pdf/2512.07448.pdf

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

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2025-12-10 04:52