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A Physics-Informed Scenario Approach with Data Mitigation for Safety Verification of Nonlinear Systems

Ali Aminzadeh, MohammadHossein Ashoori, Amy Nejati, Abolfazl Lavaei

TL;DR

This work addresses safety verification for discrete-time nonlinear systems by using barrier certificates (BCs) to guarantee infinite-horizon safety. It identifies the data burden of conventional scenario approaches and introduces a physics-informed data-selection scheme that retains only samples for which the physics-based one-step map $f^{phy}$ closely agrees with the true dynamics $f$ (i.e., $\|f^{phy}(\hat{x})-f(\hat{x})\|\le \delta$). The authors prove correctness guarantees in both deterministic and probabilistic settings, linking the SOP optimal value to the original robust optimization problem through Lipschitz continuity and beta-function bounds. Case studies on supply-demand and logistic growth demonstrate substantial data reduction while preserving safety guarantees, underscoring the method’s practical impact for real-world safety verification.

Abstract

This paper develops a physics-informed scenario approach for safety verification of nonlinear systems using barrier certificates (BCs) to ensure that system trajectories remain within safe regions over an infinite time horizon. Designing BCs often relies on an accurate dynamics model; however, such models are often imprecise due to the model complexity involved, particularly when dealing with highly nonlinear systems. In such cases, while scenario approaches effectively address the safety problem using collected data to construct a guaranteed BC for the dynamical system, they often require substantial amounts of data-sometimes millions of samples-due to exponential sample complexity. To address this, we propose a physics-informed scenario approach that selects data samples such that the outputs of the physics-based model and the observed data are sufficiently close (within a specified threshold). This approach guides the scenario optimization process to eliminate redundant samples and significantly reduce the required dataset size. We demonstrate the capability of our approach in mitigating the amount of data required for scenario optimizations with both deterministic (i.e., confidence 1) and probabilistic (i.e., confidence between 0 and 1) guarantees. We validate our physics-informed scenario approach through two physical case studies, showcasing its practical application in reducing the required data.

A Physics-Informed Scenario Approach with Data Mitigation for Safety Verification of Nonlinear Systems

TL;DR

This work addresses safety verification for discrete-time nonlinear systems by using barrier certificates (BCs) to guarantee infinite-horizon safety. It identifies the data burden of conventional scenario approaches and introduces a physics-informed data-selection scheme that retains only samples for which the physics-based one-step map closely agrees with the true dynamics (i.e., ). The authors prove correctness guarantees in both deterministic and probabilistic settings, linking the SOP optimal value to the original robust optimization problem through Lipschitz continuity and beta-function bounds. Case studies on supply-demand and logistic growth demonstrate substantial data reduction while preserving safety guarantees, underscoring the method’s practical impact for real-world safety verification.

Abstract

This paper develops a physics-informed scenario approach for safety verification of nonlinear systems using barrier certificates (BCs) to ensure that system trajectories remain within safe regions over an infinite time horizon. Designing BCs often relies on an accurate dynamics model; however, such models are often imprecise due to the model complexity involved, particularly when dealing with highly nonlinear systems. In such cases, while scenario approaches effectively address the safety problem using collected data to construct a guaranteed BC for the dynamical system, they often require substantial amounts of data-sometimes millions of samples-due to exponential sample complexity. To address this, we propose a physics-informed scenario approach that selects data samples such that the outputs of the physics-based model and the observed data are sufficiently close (within a specified threshold). This approach guides the scenario optimization process to eliminate redundant samples and significantly reduce the required dataset size. We demonstrate the capability of our approach in mitigating the amount of data required for scenario optimizations with both deterministic (i.e., confidence 1) and probabilistic (i.e., confidence between 0 and 1) guarantees. We validate our physics-informed scenario approach through two physical case studies, showcasing its practical application in reducing the required data.

Paper Structure

This paper contains 15 sections, 3 theorems, 22 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Description
  3. Notation
  4. Discrete-Time Nonlinear Systems
  5. Barrier Certificates
  6. Scenario Approach for BC Design
  7. Physics-Informed Scenario Approach for BC Design
  8. Correctness Guarantee for Physics-Informed Scenario Approach
  9. Deterministic guarantee
  10. Probabilistic guarantee
  11. Discussion
  12. Case Study
  13. Supply-Demand
  14. Logistic Growth
  15. Conclusion

Key Result

Theorem 2.2

Given a dt-NS $\Lambda = (X, f)$, suppose that $\mathcal{B}$ is a BC for $\Lambda$, as outlined in Definition BC. Then, for every initial state $x_0 \in X_0$ and any time step $k \in \mathbb{N}_0$, the state trajectory $x_{x_0}(k)$ does not enter the unsafe region $X_u$, i.e., $x_{x_0}(k) \notin X_u

Figures (3)

  • Figure 1: Physics-informed BC for the supply-demand model with (a) deterministic guarantee and (b) probabilistic guarantee. The quadratic BC is plotted for a small segment in blue, with conditions \ref{['sys2']} and \ref{['sys3']} being satisfied. Green and red dashed lines represent the initial and unsafe level sets, respectively.
  • Figure 2: Physics-informed sampling strategy according to \ref{['closedata']} for (a) supply-demand and (b) Logistic growth examples, focusing on the region where the maximum jump in sampling occurs.
  • Figure 3: Physics-informed BC for the Logistic growth model with (a) deterministic guarantee and (b) probabilistic guarantee. The quadratic BC is plotted for a small segment in blue, with conditions \ref{['sys2']} and \ref{['sys3']} being satisfied. Green and red dashed lines represent the initial and unsafe level sets, respectively.

Theorems & Definitions (7)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Remark 4.3