[Paper Review] Statistical Verification of Autonomous Systems using Surrogate Models and Conformal Inference

Chuchu Fan, Xin Qin+3 more|arXiv (Cornell University)|Apr 1, 2020

Formal Methods in Verification42 references4 citations

TL;DR

This paper proposes a conformal inference-based method for statistical verification of cyber-physical systems (CPS) using surrogate models to provide probabilistic guarantees on signal temporal logic (STL) property satisfaction. By leveraging model simulations to train surrogate models and applying conformal prediction, the approach delivers confidence intervals for robustness values and identifies safe/unsafe regions in the parameter space with user-specified confidence levels, enabling scalable, distribution-agnostic verification of safety-critical systems.

ABSTRACT

In this paper, we propose conformal inference based approach for statistical verification of CPS models. Cyber-physical systems (CPS) such as autonomous vehicles, avionic systems, and medical devices operate in highly uncertain environments. This uncertainty is typically modeled using a finite number of parameters or input signals. Given a system specification in Signal Temporal Logic (STL), we would like to verify that for all (infinite) values of the model parameters/input signals, the system satisfies its specification. Unfortunately, this problem is undecidable in general. {\em Statistical model checking} (SMC) offers a solution by providing guarantees on the correctness of CPS models by statistically reasoning on model simulations. We propose a new approach for statistical verification of CPS models for user-provided distribution on the model parameters. Our technique uses model simulations to learn {\em surrogate models}, and uses {\em conformal inference} to provide probabilistic guarantees on the satisfaction of a given STL property. Additionally, we can provide prediction intervals containing the quantitative satisfaction values of the given STL property for any user-specified confidence level. We also propose a refinement procedure based on Gaussian Process (GP)-based surrogate models for obtaining fine-grained probabilistic guarantees over sub-regions in the parameter space. This in turn enables the CPS designer to choose assured validity domains in the parameter space for safety-critical applications. Finally, we demonstrate the efficacy of our technique on several CPS models.

Motivation & Objective

• To address the challenge of verifying safety-critical CPS under uncertainty, where traditional verification is undecidable due to infinite parameter and input signal combinations.
• To provide probabilistic guarantees on STL property satisfaction across a user-defined distribution of system parameters, without assuming specific model structure or distribution.
• To enable the identification of safe and unsafe operating regions in the parameter space through refinement of surrogate models using Gaussian processes.
• To support safety assurance cases and supervisory control by generating conditional contracts based on verified parameter sub-regions.
• To overcome scalability limitations of existing verification methods by using sample-independent, non-parametric conformal prediction on simulation data.

Proposed method

• The method uses Monte Carlo sampling of system parameters from a user-provided distribution to generate model trajectories via simulation.
• For each sampled parameter, the robustness value of the corresponding trajectory with respect to a given STL formula is computed.
• A surrogate model (e.g., Gaussian Process) is trained on the parameter-robustness pairs to predict robustness values across the parameter space.
• Conformal inference is applied to the surrogate model’s predictions to generate prediction intervals with marginal coverage guarantees at a user-specified confidence level.
• A refinement procedure based on Gaussian Process regression is used to obtain tighter probabilistic bounds over sub-regions of interest in the parameter space.
• The framework supports both Boolean and quantitative STL verification, providing confidence intervals for robustness values and probability bounds on satisfaction.

Experimental results

Research questions

• RQ1 Can we provide probabilistic guarantees on the satisfaction of a given STL property across a continuous, user-provided distribution of system parameters?
• RQ2 Can we generate prediction intervals for the robustness value of an STL property that maintain a user-specified confidence level, even with limited simulation data?
• RQ3 Can we identify sub-regions in the parameter space where the system is guaranteed to satisfy or violate a specification with high probability?
• RQ4 How can we refine the surrogate model to improve the precision of probabilistic guarantees in critical sub-regions without sacrificing coverage?
• RQ5 Can the method be applied to complex, nonlinear CPS models without requiring internal model structure or assuming specific distributions?

Key findings

• The method achieved a 54.44% unsafe volume estimate for the Simglucose model in the parameter region Θ₂M, closely matching the ground truth of 52.47%.
• For the Simglucose model, the method correctly identified that the specification G[0,24] BG ≤ 155 had a 52.47% unsafe volume, while G[0,24] BG ≤ 170 had 0% unsafe volume in the same region.
• In larger parameter spaces (e.g., Θ₅M), the method estimated unsafe volumes of 6.6% for G[0,24] BG ≤ 170, demonstrating scalability to high-dimensional parameter spaces.
• The use of Gaussian Process-based refinement enabled the identification of fine-grained safe and unsafe regions, supporting conditional contracts for supervisory control.
• The conformal prediction framework maintained valid coverage guarantees regardless of the quality of the surrogate model, with wider intervals compensating for poor model fit.
• The approach is sample-independent in terms of confidence level, as conformal inference ensures coverage even with limited data, unlike traditional SMC methods requiring fixed sample size calculations.