arXiv subjects

Frederik Baymler Mathiesen

Publications and source records attributed to Frederik Baymler Mathiesen.

16 recordsLinked to original sources

arXiv2026

StochasticBarrier.jl: A Toolbox for Stochastic Barrier Function Synthesis

We present StochasticBarrier.jl, an open-source Julia-based toolbox for generating Stochastic Barrier Functions (SBFs) for safety verification of discrete-time stochastic systems with additive Gaussian noise. StochasticBarrier.jl certifies linear, polynomial, and piecewise affine (PWA) systems. The latter enables verification for a wide range of system dynamics, including general nonlinear types. The toolbox implements a Sum-of-Squares (SOS) optimization approach, as well as methods based on piecewise constant (PWC) functions. For SOS-based SBFs, StochasticBarrier.jl leverages semi-definite programming solvers, while for PWC SBFs, it offers three engines: two using linear programming (LP) and one based on gradient descent (GD). Benchmarking StochasticBarrier.jl against the state-of-the-art shows that the tool outperforms existing tools in computation time, safety probability bounds, and scalability across over 30 case studies. Compared to its closest competitor, StochasticBarrier.jl is up to four orders of magnitude faster, achieves significant safety probability improvements, and supports higher-dimensional systems.

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arXiv2026

Certified Neural Approximations of Nonlinear Dynamics

Neural networks hold great potential to act as approximate models of nonlinear dynamical systems, with the resulting neural approximations enabling verification and control of such systems. However, in safety-critical contexts, the use of neural approximations requires formal bounds on their closeness to the underlying system. To address this fundamental challenge, we propose a novel, adaptive, and parallelizable verification method based on certified first-order models. Our approach provides formal error bounds on the neural approximations of dynamical systems, allowing them to be safely employed as surrogates by interpreting the error bound as bounded disturbances acting on the approximated dynamics. We demonstrate the effectiveness and scalability of our method on a range of established benchmarks from the literature, showing that it significantly outperforms the state of the art. Furthermore, we show that our framework can successfully address additional scenarios previously intractable for existing methods -- neural network compression and an autoencoder-based deep learning architecture for training Koopman operators for the purpose of trajectory prediction.

cs.LG↗

arXiv2026

Scalable Verification of Neural Control Barrier Functions Using Linear Bound Propagation

Control barrier functions (CBFs) are a popular tool for safety certification of nonlinear dynamical control systems. Recently, CBFs represented as neural networks have shown great promise due to their expressiveness and applicability to a broad class of dynamics and safety constraints. However, verifying that a trained neural network is indeed a valid CBF is a computational bottleneck that limits the size of the networks that can be used. To overcome this limitation, we present a novel framework for verifying neural CBFs based on piecewise linear upper and lower bounds on the conditions required for a neural network to be a CBF. Our approach is rooted in linear bound propagation (LBP) for neural networks, which we extend to compute bounds on the gradients of the network. Combined with McCormick relaxation, we derive linear upper and lower bounds on the CBF conditions, thereby eliminating the need for computationally expensive verification procedures. Our approach applies to arbitrary control-affine systems and a broad range of nonlinear activation functions. To reduce conservatism, we develop a parallelizable refinement strategy that adaptively refines the regions over which these bounds are computed. Our approach scales to larger neural networks than state-of-the-art verification procedures for CBFs, as demonstrated by our numerical experiments.

cs.LG↗

arXiv2026

Evaluating randomized smoothing as a defense against adversarial attacks in trajectory prediction

Accurate and robust trajectory prediction is essential for safe and efficient autonomous driving, yet recent work has shown that even state-of-the-art prediction models are highly vulnerable to inputs being mildly perturbed by adversarial attacks. Although model vulnerabilities to such attacks have been studied, work on effective countermeasures remains limited. In this work, we develop and evaluate a new defense mechanism for trajectory prediction models based on randomized smoothing -- an approach previously applied successfully in other domains. We evaluate its ability to improve model robustness through a series of experiments that test different strategies of randomized smoothing. We show that our approach can consistently improve prediction robustness of multiple base trajectory prediction models in various datasets without compromising accuracy in non-adversarial settings. Our results demonstrate that randomized smoothing offers a simple and computationally inexpensive technique for mitigating adversarial attacks in trajectory prediction.

cs.LG↗

arXiv2025

ARCH-COMP25 Category Report: Stochastic Models

This report is concerned with a friendly competition for formal verification and policy synthesis of stochastic models. The main goal of the report is to introduce new benchmarks and their properties within this category and recommend next steps toward next year's edition of the competition. In particular, this report introduces three recently developed software tools, a new water distribution network benchmark, and a collection of simplified benchmarks intended to facilitate further comparisons among tools that were previously not directly comparable. This friendly competition took place as part of the workshop Applied Verification for Continuous and Hybrid Systems (ARCH) in Summer 2025.

cs.LO↗

arXiv2025

Realistic Adversarial Attacks for Robustness Evaluation of Trajectory Prediction Models via Future State Perturbation

Trajectory prediction is a key element of autonomous vehicle systems, enabling them to anticipate and react to the movements of other road users. Evaluating the robustness of prediction models against adversarial attacks is essential to ensure their reliability in real-world traffic. However, current approaches tend to focus on perturbing the past positions of surrounding agents, which can generate unrealistic scenarios and overlook critical vulnerabilities. This limitation may result in overly optimistic assessments of model performance in real-world conditions. In this work, we demonstrate that perturbing not just past but also future states of adversarial agents can uncover previously undetected weaknesses and thereby provide a more rigorous evaluation of model robustness. Our novel approach incorporates dynamic constraints and preserves tactical behaviors, enabling more effective and realistic adversarial attacks. We introduce new performance measures to assess the realism and impact of these adversarial trajectories. Testing our method on a state-of-the-art prediction model revealed significant increases in prediction errors and collision rates under adversarial conditions. Qualitative analysis further showed that our attacks can expose critical weaknesses, such as the inability of the model to detect potential collisions in what appear to be safe predictions. These results underscore the need for more comprehensive adversarial testing to better evaluate and improve the reliability of trajectory prediction models for autonomous vehicles.

cs.LG↗

arXiv2025

Scalable control synthesis for stochastic systems via structural IMDP abstractions

This paper introduces a novel abstraction-based framework for controller synthesis of nonlinear discrete-time stochastic systems. The focus is on probabilistic reach-avoid specifications. The framework is based on abstracting a stochastic system into a new class of robust Markov models, called orthogonally decoupled Interval Markov Decision Processes (odIMDPs). Specifically, an odIMDPs is a class of robust Markov processes, where the transition probabilities between each pair of states are uncertain and have the product form. We show that such a specific form in the transition probabilities allows one to build compositional abstractions of stochastic systems that, for each state, are only required to store the marginal probability bounds of the original system. This leads to improved memory complexity for our approach compared to commonly employed abstraction-based approaches. Furthermore, we show that an optimal control strategy for a odIMDPs can be computed by solving a set of linear problems. When the resulting strategy is mapped back to the original system, it is guaranteed to lead to reduced conservatism compared to existing approaches. To test our theoretical framework, we perform an extensive empirical comparison of our methods against Interval Markov Decision Process- and Markov Decision Process-based approaches on various benchmarks including 7D systems. Our empirical analysis shows that our approach substantially outperforms state-of-the-art approaches in terms of both memory requirements and the conservatism of the results.

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arXiv2024

Piecewise Stochastic Barrier Functions

This paper presents a novel stochastic barrier function (SBF) framework for safety analysis of stochastic systems based on piecewise (PW) functions. We first outline a general formulation of PW-SBFs. Then, we focus on PW-Constant (PWC) SBFs and show how their simplicity yields computational advantages for general stochastic systems. Specifically, we prove that synthesis of PWC-SBFs reduces to a minimax optimization problem. Then, we introduce three efficient algorithms to solve this problem, each offering distinct advantages and disadvantages. The first algorithm is based on dual linear programming (LP), which provides an exact solution to the minimax optimization problem. The second is a more scalable algorithm based on iterative counter-example guided synthesis, which involves solving two smaller LPs. The third algorithm solves the minimax problem using gradient descent, which admits even better scalability. We provide an extensive evaluation of these methods on various case studies, including neural network dynamic models, nonlinear switched systems, and high-dimensional linear systems. Our benchmarks demonstrate that PWC-SBFs outperform state-of-the-art methods, namely sum-of-squares and neural barrier functions, and can scale to eight dimensional systems.

cs.RO↗

arXiv2024

A data-driven approach for safety quantification of non-linear stochastic systems with unknown additive noise distribution

In this paper, we present a novel data-driven approach to quantify safety for non-linear, discrete-time stochastic systems with unknown noise distribution. We define safety as the probability that the system remains in a given region of the state space for a given time horizon and, to quantify it, we present an approach based on Stochastic Barrier Functions (SBFs). In particular, we introduce an inner approximation of the stochastic program to design a SBF in terms of a chance-constrained optimisation problem, which allows us to leverage the scenario approach theory to design a SBF from samples of the system with Probably Approximately Correct (PAC) guarantees. Our approach leads to tractable, robust linear programs, which enable us to assert safety for non-linear models that were otherwise deemed infeasible with existing methods. To further mitigate the computational complexity of our approach, we exploit the structure of the system dynamics and rely on spatial data structures to accelerate the construction and solution of the underlying optimisation problem. We show the efficacy and validity of our framework in several benchmarks, showing that our approach can obtain substantially tighter certificates compared to state-of-the-art with a confidence that is several orders of magnitude higher.

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arXiv2024

Data-Driven Permissible Safe Control with Barrier Certificates

This paper introduces a method of identifying a maximal set of safe strategies from data for stochastic systems with unknown dynamics using barrier certificates. The first step is learning the dynamics of the system via Gaussian process (GP) regression and obtaining probabilistic errors for this estimate. Then, we develop an algorithm for constructing piecewise stochastic barrier functions to find a maximal permissible strategy set using the learned GP model, which is based on sequentially pruning the worst controls until a maximal set is identified. The permissible strategies are guaranteed to maintain probabilistic safety for the true system. This is especially important for learning-enabled systems, because a rich strategy space enables additional data collection and complex behaviors while remaining safe. Case studies on linear and nonlinear systems demonstrate that increasing the size of the dataset for learning the system grows the permissible strategy set.

cs.LG↗

arXiv2024

IntervalMDP.jl: Accelerated Value Iteration for Interval Markov Decision Processes

In this paper, we present IntervalMDP.jl, a Julia package for probabilistic analysis of interval Markov Decision Processes (IMDPs). IntervalMDP.jl facilitates the synthesis of optimal strategies and verification of IMDPs against reachability specifications and discounted reward properties. The library supports sparse matrices and is compatible with data formats from common tools for the analysis of probabilistic models, such as PRISM. A key feature of IntervalMDP.jl is that it presents both a multi-threaded CPU and a GPU-accelerated implementation of value iteration algorithms for IMDPs. In particular, IntervalMDP.jl takes advantage of the Julia type system and the inherently parallelizable nature of value iteration to improve the efficiency of performing analysis of IMDPs. On a set of examples, we show that IntervalMDP.jl substantially outperforms existing tools for verification and strategy synthesis for IMDPs in both computation time and memory consumption.

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arXiv2024

A survey on robustness in trajectory prediction for autonomous vehicles

Autonomous vehicles rely on accurate trajectory prediction to inform decision-making processes related to navigation and collision avoidance. However, current trajectory prediction models show signs of overfitting, which may lead to unsafe or suboptimal behavior. To address these challenges, this paper presents a comprehensive framework that categorizes and assesses the definitions and strategies used in the literature on evaluating and improving the robustness of trajectory prediction models. This involves a detailed exploration of various approaches, including data slicing methods, perturbation techniques, model architecture changes, and post-training adjustments. In the literature, we see many promising methods for increasing robustness, which are necessary for safe and reliable autonomous driving.

cs.RO↗

arXiv2023

Simultaneous Synthesis and Verification of Neural Control Barrier Functions through Branch-and-Bound Verification-in-the-loop Training

Control Barrier Functions (CBFs) that provide formal safety guarantees have been widely used for safety-critical systems. However, it is non-trivial to design a CBF. Utilizing neural networks as CBFs has shown great success, but it necessitates their certification as CBFs. In this work, we leverage bound propagation techniques and the Branch-and-Bound scheme to efficiently verify that a neural network satisfies the conditions to be a CBF over the continuous state space. To accelerate training, we further present a framework that embeds the verification scheme into the training loop to synthesize and verify a neural CBF simultaneously. In particular, we employ the verification scheme to identify partitions of the state space that are not guaranteed to satisfy the CBF conditions and expand the training dataset by incorporating additional data from these partitions. The neural network is then optimized using the augmented dataset to meet the CBF conditions. We show that for a non-linear control-affine system, our framework can efficiently certify a neural network as a CBF and render a larger safe set than state-of-the-art neural CBF works. We further employ our learned neural CBF to derive a safe controller to illustrate the practical use of our framework.

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arXiv2023

Inner approximations of stochastic programs for data-driven stochastic barrier function design

This paper proposes a new framework to compute finite-horizon safety guarantees for discrete-time piece-wise affine systems with stochastic noise of unknown distributions. The approach is based on a novel approach to synthesise a stochastic barrier function (SBF) from noisy data and rely on the scenario optimization theory. In particular, we show that the stochastic program to synthesize a SBF can be relaxed into a chance-constrained optimisation problem on which scenario approach theory applies. We further show that the resulting program can be reduced to a linear programming problem, thus guaranteeing efficiency. In contrast to existing approaches, this method is data efficient as it only requires the number of data to be proportional to the logarithm in the negative inverse of the confidence level and is computationally efficient due to its reduction to linear programming. The efficacy of the method is empirically evaluated on various verification benchmarks. Experiments show a significant improvement with respect to state-of-the-art, obtaining tighter certificates with a confidence that is several orders of magnitude higher.

eess.SY↗

arXiv2022

Safety Certification for Stochastic Systems via Neural Barrier Functions

Providing non-trivial certificates of safety for non-linear stochastic systems is an important open problem that limits the wider adoption of autonomous systems in safety-critical applications. One promising solution to address this problem is barrier functions. The composition of a barrier function with a stochastic system forms a supermartingale, thus enabling the computation of the probability that the system stays in a safe set over a finite time horizon via martingale inequalities. However, existing approaches to find barrier functions for stochastic systems generally rely on convex optimization programs that restrict the search of a barrier to a small class of functions such as low degree SoS polynomials and can be computationally expensive. In this paper, we parameterize a barrier function as a neural network and show that techniques for robust training of neural networks can be successfully employed to find neural barrier functions. Specifically, we leverage bound propagation techniques to certify that a neural network satisfies the conditions to be a barrier function via linear programming and then employ the resulting bounds at training time to enforce the satisfaction of these conditions. We also present a branch-and-bound scheme that makes the certification framework scalable. We show that our approach outperforms existing methods in several case studies and often returns certificates of safety that are orders of magnitude larger.

eess.SY↗

arXiv2020

A Flow-Efficient and Legal-by-Construction Real-Time Traffic Signal Control Platform

Inefficiencies in traffic flow through an intersection lead to stopping vehicles, unnecessary congestion, and increased accident risk. In this paper, we propose a traffic signal controller platform demonstrating the ability to increase traffic flow for arbitrary intersection topologies. This model uses Model Predictive Control on a Mixed Logical Dynamical system to control the state of independently controlled traffic signals in a single intersection, removing constraints forcing the selection of signals from a set of phases. Further, we use constraints to impose a guarantee on the output of the system to be in the set of permissible actions under constraints including precise yellow timing, minimum inter-lane green transition timing, and selection of signal states with non-conflicting dependencies. We evaluate our model on a simulated 4-way intersection and an intersection in Denmark with true traffic data and a currently implemented timing schedule as a baseline. Our model shows at least 22\% reduction in time loss compared to baseline light schedules, and timing shows that this system can feasibly run online predictions at a frequency faster than 2s/prediction optimizing over a prediction horizon of 25s.

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