We study the problem of certifying the robustness of Bayesian neural networks (BNNs) to adversarial input perturbations. Specifically, we define two notions of robustness for BNNs in an adversarial setting: probabilistic robustness and decision robustness. The former deals with the probabilistic behaviour of the network, that is, it ensures robustness across different stochastic realisations of the network, while the latter provides guarantees for the overall (output) decision of the BNN. Although these robustness properties cannot be computed analytically, we present a unified computational framework for efficiently and formally bounding them. Our approach is based on weight interval sampling, integration and bound propagation techniques, and can be applied to BNNs with a large number of parameters independently of the (approximate) inference method employed to train the BNN. We evaluate the effectiveness of our method on tasks including airborne collision avoidance, medical imaging and autonomous driving, demonstrating that it can compute non-trivial guarantees on medium size images (i.e., over 16 thousand input parameters).

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We introduce a framework for the control of discrete-time switched stochastic systems with uncertain distributions. In particular, we consider stochastic dynamics with additive noise whose distribution lies in an ambiguity set of distributions that are ɛ−close, in the Wasserstein distance sense, to a nominal one. We propose algorithms for the efficient synthesis of distributionally robust control strategies that maximize the satisfaction probability of reach-avoid specifications with either a given or an arbitrary (not specified) time horizon, i.e., unbounded-time reachability. The framework consists of two main steps: finite abstraction and control synthesis. First, we construct a finite abstraction of the switched stochastic system as a robust Markov decision process (robust MDP) that encompasses both the stochasticity of the system and the uncertainty in the noise distribution. Then, we synthesize a strategy that is robust to the distributional uncertainty on the resulting robust MDP. We employ techniques from optimal transport and stochastic programming to reduce the strategy synthesis problem to a set of linear programs, and propose a tailored and efficient algorithm to solve them. The resulting strategies are correctly refined into switching strategies for the original stochastic system. We illustrate the efficacy of our framework on various case studies comprising both linear and non-linear switched stochastic systems.@en

In this study, we explore the mechanisms underlying the exceptional intrinsic strength of face-centered cubic (FCC) Multi-Principal Element Alloys (MPEAs) using a multifaceted approach. Our methods integrate atomistic simulations, informed by both embedded-atom model and neural network potentials, with first-principles calculations, stochastic Peierls-Nabarro (PN) modeling, and symbolic machine learning. We identify a consistent, robust linear correlation between the strength of MPEAs and the standard deviation of the maximum stacking-fault restoring force (τ_max,sd) across various potentials. This finding is substantiated by comparing the experimental strengths of Cantor alloys’ subsystems and Ni_62.5V_37.5 against τ_max,sd values from high-throughput first-principle calculations. Our theoretical insights are derived from integrating the stochastic Peierls-Nabarro model with a shearable precipitation hardening framework, demonstrating that lattice distortion alone does not directly enhance intrinsic strength. Instead, τ_max,sd emerges as a critical determinant, capable of boosting the strength of MPEAs by up to tenfold. Our analysis reveals the critical role of the exponential form of the PN model in achieving substantial strength improvement by transforming the Gaussian-like distribution of τ_max into an exponential-like distribution of local Peierls stress. Additionally, using an advanced symbolic machine learning technique, the sure independence screening and sparsifying operator (SISSO) method, we derive interpretable relationships between MPEA strength, elastic properties, and τ_max statistics, offering new insights into the design and optimization of advanced MPEAs. These findings highlight that the nonlinear physics and atomic fluctuations characterizing MPEAs not only underpin their unconventional intrinsic strength but also contribute to other complex properties such as sluggish diffusion and cocktail effect.

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Vulnerability to adversarial attacks is one of the principal hurdles to the adoption of deep learning in safety-critical applications. Despite significant efforts, both practical and theoretical, training deep learning models robust to adversarial attacks is still an open problem. In this article, we analyse the geometry of adversarial attacks in the over-parameterized limit for Bayesian neural networks (BNNs). We show that, in the limit, vulnerability to gradient-based attacks arises as a result of degeneracy in the data distribution, i.e., when the data lie on a lower dimensional submanifold of the ambient space. As a direct consequence, we demonstrate that in this limit, BNN posteriors are robust to gradient-based adversarial attacks. Crucially, by relying on the convergence of infinitely-wide BNNs to Gaussian processes (GPs), we prove that, under certain relatively mild assumptions, the expected gradient of the loss with respect to the BNN posterior distribution is vanishing, even when each NN sampled from the BNN posterior does not have vanishing gradients. The experimental results on the MNIST, Fashion MNIST, and a synthetic dataset with BNNs trained with Hamiltonian Monte Carlo and variational inference support this line of arguments, empirically showing that BNNs can display both high accuracy on clean data and robustness to both gradient-based and gradient-free adversarial attacks.

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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.@en

Accurate fatigue assessment of material plagued by defects is of utmost importance to guarantee safety and service continuity in engineering components. This study shows how state-of-the-art semi-empirical models can be endowed with additional defect descriptors to probabilistically predict the occurrence of fatigue failures by exploiting advanced Bayesian Physics-guided Neural Network (B-PGNN) approaches. A B-PGNN is thereby developed to predict the fatigue failure probability of a sample containing defects, referred to a given fatigue endurance limit. In this framework, a robustly calibrated El Haddad's curve is exploited as the prior physics reinforcement of the probabilistic model, i.e., prior knowledge. Following, a likelihood function is built and the B-PGNN is trained via Bayesian Inference, thus calculating the posterior of the parameters. The arbitrariness of the choice of the related architecture is circumvented through a Bayesian model selection strategy. A case-study is analysed to prove the robustness of the proposed approach. This methodology proposes an advanced practical approach to help support the probabilistic design against fatigue failure.

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Model-based reinforcement learning seeks to simultaneously learn the dynamics of an unknown stochastic environment and synthesise an optimal policy for acting in it. Ensuring the safety and robustness of sequential decisions made through a policy in such an environment is a key challenge for policies intended for safety-critical scenarios. In this work, we investigate two complementary problems: first, computing reach-avoid probabilities for iterative predictions made with dynamical models, with dynamics described by Bayesian neural network (BNN); second, synthesising control policies that are optimal with respect to a given reach-avoid specification (reaching a “target” state, while avoiding a set of “unsafe” states) and a learned BNN model. Our solution leverages interval propagation and backward recursion techniques to compute lower bounds for the probability that a policy's sequence of actions leads to satisfying the reach-avoid specification. Such computed lower bounds provide safety certification for the given policy and BNN model. We then introduce control synthesis algorithms to derive policies maximizing said lower bounds on the safety probability. We demonstrate the effectiveness of our method on a series of control benchmarks characterized by learned BNN dynamics models. On our most challenging benchmark, compared to purely data-driven policies the optimal synthesis algorithm is able to provide more than a four-fold increase in the number of certifiable states and more than a three-fold increase in the average guaranteed reach-avoid probability.

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Analyzing event-triggered control's (ETC) sampling behavior is of paramount importance, as it enables formal assessment of its sampling performance and prediction of its sampling patterns. In this work, we formally analyze the sampling behavior of stochastic linear periodic ETC (PETC) systems by computing bounds on associated metrics. Specifically, we consider functions over sequences of state measurements and intersampling times that can be expressed as average, multiplicative or cumulative rewards, and introduce their expectations as metrics on PETC's sampling behavior. We compute bounds on these expectations, by constructing Interval Markov Chains equipped with suitable reward functions, that abstract stochastic PETC's sampling behavior. Our results are illustrated on a numerical example, for which we compute bounds on the expected average intersampling time and on the probability of triggering with the maximum possible intersampling time in a finite horizon.

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We present a novel framework for formal control of uncertain discrete-time switched stochastic systems against probabilistic reach-avoid specifications. In particular, we consider stochastic systems with additive noise, whose distribution lies in an ambiguity set of distributions that are ε−close to a nominal one according to the Wasserstein distance. For this class of systems we derive control synthesis algorithms that are robust against all these distributions and maximize the probability of satisfying a reach-avoid specification, defined as the probability of reaching a goal region while being safe. The framework we present first learns an abstraction of a switched stochastic system as a robust Markov decision process (robust MDP) by accounting for both the stochasticity of the system and the uncertainty in the noise distribution. Then, it synthesizes a strategy on the resulting robust MDP that maximizes the probability of satisfying the property and is robust to all uncertainty in the system. This strategy is then refined into a switching strategy for the original stochastic system. By exploiting tools from optimal transport and stochastic programming, we show that synthesizing such a strategy reduces to solving a set of linear programs, thus guaranteeing efficiency. We experimentally validate the efficacy of our framework on various case studies, including both linear and non-linear switched stochastic systems. Our results represent the first formal approach for control synthesis of stochastic systems with uncertain noise distribution.

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Providing non-trivial certificates of safety for non-linear stochastic systems is an important open problem. One promising solution to address this problem is the use of barrier functions. Barrier functions are functions whose composition with the system forms a Martingale and enable the computation of the probability that the system stays within a safe set over a finite time horizon. However, existing approaches to find barrier functions generally restrict the search to a small class of functions, often leading to conservatism. To address this problem, in this letter, we parameterize barrier functions as neural networks and show that bound propagation techniques and linear programming can be successfully employed to find Neural Barrier Functions. Further, we develop a branch-and-bound scheme based on linear relaxations that improves the scalability of the proposed framework. On several case studies we show that our approach scales to neural networks of hundreds of neurons and multiple hidden layers and often produces certificates of safety that are tighter than state-of-the-art methods.@en

Interval Markov Decision Processes (IMDPs) are finite-state uncertain Markov models, where the transition probabilities belong to intervals. Recently, there has been a surge of research on employing IMDPs as abstractions of stochastic systems for control synthesis. However, due to the absence of algorithms for synthesis over IMDPs with continuous action-spaces, the action-space is assumed discrete a-priori, which is a restrictive assumption for many applications. Motivated by this, we introduce continuous-action IMDPs (caIMDPs), where the bounds on transition probabilities are functions of the action variables, and study value iteration for maximizing expected cumulative rewards. Specifically, we decompose the max-min problem associated to value iteration to |Q| max problems, where |Q| is the number of states of the caIMDP. Then, exploiting the simple form of these max problems, we identify cases where value iteration over caIMDPs can be solved efficiently (e.g., with linear or convex programming). We also gain other interesting insights: e.g., in certain cases where the action set A is a polytope, synthesis over a discrete-action IMDP, where the actions are the vertices of A, is sufficient for optimality. We demonstrate our results on a numerical example. Finally, we include a short discussion on employing caIMDPs as abstractions for control synthesis.

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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. @en

Verifying the performance of safety-critical, stochastic systems with complex noise distributions is difficult. We introduce a general procedure for the finite abstraction of nonlinear stochastic systems with nonstandard (e.g., non-affine, non-symmetric, non-unimodal) noise distributions for verification purposes. The method uses a finite partitioning of the noise domain to construct an interval Markov chain (IMC) abstraction of the system via transition probability intervals. Noise partitioning allows for a general class of distributions and structures, including multiplicative and mixture models, and admits both known and data-driven systems. The partitions required for optimal transition bounds are specified for systems that are monotonic with respect to the noise, and explicit partitions are provided for affine and multiplicative structures. By the soundness of the abstraction procedure, verification on the IMC provides guarantees on the stochastic system against a temporal logic specification. In addition, we present a novel refinement-free algorithm that improves the verification results. Case studies on linear and nonlinear systems with non-Gaussian noise, including a data-driven example, demonstrate the generality and effectiveness of the method without introducing excessive conservatism.

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Deep Kernel Learning (DKL) combines the representational power of neural networks with the uncertainty quantification of Gaussian Processes. Hence, it is potentially a promising tool to learn and control complex dynamical systems. In this letter, we develop a scalable abstraction-based framework that enables the use of DKL for control synthesis of stochastic dynamical systems against complex specifications. Specifically, we consider temporal logic specifications and create an end-to-end framework that uses DKL to learn an unknown system from data and formally abstracts the DKL model into an interval Markov decision process to perform control synthesis with correctness guarantees. Furthermore, we identify a deep architecture that enables accurate learning and efficient abstraction computation. The effectiveness of our approach is illustrated on various benchmarks, including a 5-D nonlinear stochastic system, showing how control synthesis with DKL can substantially outperform state-of-the-art competitive methods.

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In this paper, we introduce BNN-DP, an efficient algorithmic framework for analysis of adversarial robustness of Bayesian Neural Networks (BNNs). Given a compact set of input points T ⊂ Rⁿ, BNN-DP computes lower and upper bounds on the BNN's predictions for all the points in T. The framework is based on an interpretation of BNNs as stochastic dynamical systems, which enables the use of Dynamic Programming (DP) algorithms to bound the prediction range along the layers of the network. Specifically, the method uses bound propagation techniques and convex relaxations to derive a backward recursion procedure to over-approximate the prediction range of the BNN with piecewise affine functions. The algorithm is general and can handle both regression and classification tasks. On a set of experiments on various regression and classification tasks and BNN architectures, we show that BNN-DP outperforms state-of-the-art methods by up to four orders of magnitude in both tightness of the bounds and computational efficiency.

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Analyzing Event-Triggered Control's (ETC) sampling behaviour is of paramount importance, as it enables formal assessment of its sampling performance and prediction of its sampling patterns. In this work, we formally analyze the sampling behaviour of stochastic linear periodic ETC (PETC) systems by computing bounds on associated metrics. Specifically, we consider functions over sequences of state measurements and intersampling times that can be expressed as average, multiplicative or cumulative rewards, and introduce their expectations as metrics on PETC's sampling behaviour. We compute bounds on these expectations, by constructing appropriate Interval Markov Chains equipped with suitable reward structures, that abstract stochastic PETC's sampling behaviour. Our results are illustrated on a numerical example, for which we compute bounds on the expected average intersampling time and on the probability of triggering with the maximum possible intersampling time in a finite horizon.@en

Background: Over the past 20 years, the Contour Method (CM) has been extensively implemented to evaluate residual stress at the macro scale, especially in products where material processing is involved. Despite this, insufficient attention has been devoted to addressing the problems of input data filtering and residual stress uncertainties quantification. Objective: The present research aims to tackle this fundamental issue by combining Gaussian Process Regression (GPR) with the CM. Thanks to its stochastic nature, GPR associates a Gaussian distribution with every subset of data, thus holding the potential to model the inherent uncertainty of the input data set and to link it to the measurements and the surface roughness. Methods: The conventional and unrobust spline smoothing process is effectively replaced by the GPR which is capable of providing uncertainties over the fitting. Indeed, the GPR stochastically and automatically identifies the fitting parameter, thus making the experimental data post-processing practically unaffected by the user’s experience. Moreover, the final residual stress uncertainty is efficiently evaluated through an optimised Monte Carlo Finite Element simulation, by appropriately perturbing the input dataset according to the GPR predictions. Results: The simulation is globally optimised exploiting numerical techniques, such as LU-factorisation, and developing an on-line convergence criterion. In order to show the capability of the presented approach, a Friction Stir Welded plate is considered as a case study. For this problem, it was shown how residual stress and its uncertainty can be accurately evaluated in approximately 15 minutes using a low-budget personal computer. Conclusions: The method developed herein overcomes the key limitation of the standard spline smoothing approach and this provides a robust and optimised computational framework for routinely evaluating the residual stress and its associated uncertainty. The implications are very significant as the evaluation accuracy of the CM is now taken to a higher level.

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Neural networks (NNs) are emerging as powerful tools to represent the dynamics of control systems with complicated physics or black-box components. Due to complexity of NNs, however, existing methods are unable to synthesize complex behaviors with guarantees for NN dynamic models (NNDMs). This letter introduces a control synthesis framework for stochastic NNDMs with performance guarantees. The focus is on specifications expressed in linear temporal logic interpreted over finite traces (LTLf), and the approach is based on finite abstraction. Specifically, we leverage recent techniques for convex relaxation of NNs to formally abstract a NNDM into an interval Markov decision process (IMDP). Then, a strategy that maximizes the probability of satisfying a given specification is synthesized over the IMDP and mapped back to the underlying NNDM. We show that the process of abstracting NNDMs to IMDPs reduces to a set of convex optimization problems, hence guaranteeing efficiency. We also present an adaptive refinement procedure that makes the framework scalable. On several case studies, we illustrate that our framework is able to provide non-trivial guarantees of correctness for NNDMs with architectures of up to 5 hidden layers and hundreds of neurons per layer.@en

We consider the problem of certifying the individual fairness (IF) of feed-forward neural networks (NNs). In particular, we work with the ϵ-δ-IF formulation, which, given a NN and a similarity metric learnt from data, requires that the output difference between any pair of ϵ-similar individuals is bounded by a maximum decision tolerance δ ≥ 0. Working with a range of metrics, including the Mahalanobis distance, we propose a method to over-approximate the resulting optimisation problem using piecewise-linear functions to lower and upper bound the NN's non-linearities globally over the input space. We encode this computation as the solution of a Mixed-Integer Linear Programming problem and demonstrate that it can be used to compute IF guarantees on four datasets widely used for fairness benchmarking. We show how this formulation can be used to encourage models' fairness at training time by modifying the NN loss, and empirically confirm our approach yields NNs that are orders of magnitude fairer than state-of-the-art methods.

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Defects in additively manufactured materials are one of the leading sources of uncertainty in mechanical fatigue. Fracture mechanics concepts are useful to evaluate their influence, nevertheless, these approaches cannot account for the real morphology of defects. Preliminary attempts to exploit a more comprehensive description of defects can be found in the literature, by using Machine Learning. These approaches are notoriously data-hungry and neither physics laws nor phenomenological rules are introduced to assess the soundness of the outcome. Hereby, to overcome this limitation, an approach to predicting fatigue finite life of defective materials, based on a Physics-Informed Neural Network framework, is presented for the first time. The training process of a Neural Network is reinforced by introducing novel Fracture Mechanics constraints. Experimental results obtained from the literature, including detailed defect analysis from computer tomography and fractography, were used to check its accuracy. The proposed predictive tool fully exploits the advanced capabilities of machine learning to account for morphological aspects of defects that could not be accounted for otherwise, while at the same time obeying fracture mechanics laws and requiring a smaller experimental dataset. The approach paves the way for new structural design approaches with an unprecedented degree of accuracy.

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