Verification Of Neural Networks Research Articles

Reachability in Simple Neural Networks

We investigate the complexity of the reachability problem for (deep) neural networks: does it compute valid output given some valid input? It was recently claimed that the problem is NP-complete for general neural networks and specifications over the input/output dimension given by conjunctions of linear inequalities. We recapitulate the proof and repair some flaws in the original upper and lower bound proofs. Motivated by the general result, we show that NP-hardness already holds for restricted classes of simple specifications and neural networks. Allowing for a single hidden layer and an output dimension of one as well as neural networks with just one negative, zero and one positive weight or bias is sufficient to ensure NP-hardness. Additionally, we give a thorough discussion and outlook of possible extensions for this direction of research on neural network verification.

Based on YOLO v3 Target Recognition Algorithm as Vehicle Tracking Algorithm Analysis

Traditional traffic information acquisition and acquisition are mainly implemented by sensors, and these traditional acquisition and acquisition systems have some great drawbacks. However, with the popularity of traffic monitoring, computer vision technology gradually has a platform foundation that can be applied to identify and track traffic conditions. In this paper, through the research of traditional and Deep learning-based multi-target recognition algorithms and two common multi-target tracking algorithms, a solution of YOLO v3 network combined with deep-sort algorithm is proposed. In this paper, a video of traffic information of urban roads is directly collected for areas with relatively large traffic flow. Interval frames are extracted from the video data set to make relevant data sets for training and verification of YOLO v3 neural networks. Combined with the test results, an open source vehicle depth model dataset is used to train the vehicle depth feature weight file, and Deep-SORT algorithm is used to achieve the target tracking, which can realize the real-time and more accurate multi-target recognition and tracking of moving vehicles.

Model Based Verification of Spiking Neural Networks in Cyber Physical Systems

Spiking Neural Networks (SNNs) have found increasing utility in designing safety-critical Cyber-Physical Systems (CPSs) such as implantable medical devices, autonomous vehicles, and space robotics due to their capability to operate on information represented in temporal coding and exhibit various behavioural modalities. Thus, there has been recent interest in formally verifying their timing behaviours and providing soundness guarantees of their diverse characteristics. However, beyond the simplistic Leaky Integrate and Fire (LIF) model, which only mimics 3 spiking behaviours, there is a lack of unifying methodology in literature to verify complex dynamics of biological neurons exhibiting 20 spiking behaviours as demonstrated by the pioneering work of Izhikevich. There is also a complete lack of formulation for the verification of SNN-based systems. This paper bridges these gaps by proposing a model-based approach for designing SNN-based controllers in CPS. We propose sound structural transformations translating any spiking neuron into networks of Timed Automata (TA), model the complex Izhikevich neural model and formally verify all 20 timing behaviours it exhibits for the first time. We then present two case studies that were modelled as SNNs using our approach: the PID controller, and the Car-Following controller, and subsequently attempt static model checking and statistical verification of their generated TA models for safety guarantees.

Sparse polynomial optimisation for neural network verification

The prevalence of neural networks in applications is expanding at an increasing rate. It is becoming clear that providing robust guarantees on systems that use neural networks is very important, especially in safety-critical applications. A trained neural network’s sensitivity to adversarial attacks is one of its greatest shortcomings. To provide robust guarantees, one popular method that has seen success is to describe and bound the activation functions in the neural network using equality and inequality constraints. However, there are numerous ways to form these bounds, providing a trade-off between conservativeness and complexity. We approach the problem from a different perspective, using sparse polynomial optimisation theory and Positivstellensatz, a key result in real algebraic geometry. The former exploits the natural cascading structure of the neural network using ideas from chordal sparsity while the latter tests the emptiness of a semi-algebraic set using algebra, to provide tight bounds. We show that bounds can be tightened significantly, whilst the computational time remains reasonable. We compare the solve times of different solvers and show how accuracy can be improved at the expense of increased computation time. In particular, we show that with the sparse polynomial framework the solve time and accuracy can be improved over other methods for neural network verification, for networks with ReLU, sigmoid and tanh activation functions.

Research of Flight Safety and Technical Evaluation

Flight safety is the foundation of survival and development of civil aviation transport industry. Aircraft flight safety can be checked by aircraft flight parameters. We discuss a flight technology evaluation method based on flight parameter data. Firstly, data preprocessing is carried out, and the data is reduced by PCA dimension reduction. Then, BP neural network and decision tree model are established to train and test the data set. Neural network algorithm and machine learning algorithm are used for training. The results show that the BP neural network verification set is more accurate. The comparison table between the predicted results and the real results is shown in Table 3. Among the 15 samples, 14 are correct. The prediction accuracy reaches 93%.

A Study on Perception of Visual–Tactile and Color–Texture Features of Footwear Leather for Symmetric Shoes

The study applies Kansei engineering in analyzing the color and texture of leather footwear, utilizing neural network verification to mirror consumers’ visual and tactile imageries onto varieties of leather. This aids in the development of an advanced system for selecting leather footwear based on such impressions. Initially, representative word pairs denoting consumers’ visual and tactile perceptions of leather footwear were delineated. Post-evaluation of these perceptions through a sensibility assessment questionnaire was administered, using 54 samples of leather footwear provided by manufacturers, with each leather type codified in terms of visual and tactile sensibilities. Subsequently, a customized software algorithm was crafted to isolate the primary color and adhesiveness as color features from the leather sample images. Analyzing grayscale values of the images and using pixel neighborhood as a base, the associated calculation methods, such as LBP, SCOV, VAR, SAC, etc., were proposed to extract texture features from the images. The derived color and texture feature values were used as the input layer and the sensory vocabulary quantified values as the output layer. Backpropagation neural network training was conducted on 49 leather samples, with five leather samples used for testing, culminating in the verification of neural network training for three types and 17 combinations. The outcome was an optimal method for leather footwear Kansei engineering and neural network training, establishing a design process for leather footwear characteristics assisted by sensory vocabulary and a backpropagation neural network. Additionally, a computer-aided system for selecting leather footwear, based on these impressions, was designed and validated through footwear design. This study utilized symmetry in footwear design. By using the design of a single shoe to represent the imagery of a pair of symmetrical shoes, we verified whether the leather samples recommended by the leather imagery selection query system met the expected system input settings.

Intelligent system for analyzing battery charge consumption processes

The article develops an intelligent system of analysis and neural network forecasting of battery charge consumption for automated vehicles (AGVs). For this purpose, the types of AGV and the methods of effective forecasting of their battery charge consumption were analyzed. It is established that they are based on optimal robot control processes; application of technologies to increase capacity and extend service life. The data for the forecast was collected using the UAExpert OPC UA client, which allowed to convert the informative components of the data vector into a format suitable for further processing (csv). To eliminate outliers in the signals, a dispersion analysis of each parameter of AGV was carried out. Data for which the sigma value exceeded 1.5 were considered partialle lost and were replaced by a moving average of 12 points (the number of ANN inputs). For training, verification and testing of neural networks, parameters with high and medium positive correlation dependence were selected according to the Pearson correlation coefficient. Short-term and medium-term forecasting of battery charge consumption for AKTZ was carried out on the basis of ANN with deep learning, the model of which was tested in two modes: forecasting and prediction. The effectiveness of the developed system was investigated by testing it on the data obtained from Formica-1 AGV. The average absolute testing error was less than 1%. The highest value of the prediction error was less than 9 % when predicting such parameters as current position and X-coordinate, which are correlated with battery charge consumption for AGV. It has been established experimentally that the accuracy of the forecast of battery charge consumption for various types of AGV has been improved.

Leveraging Satisfiability Modulo Theory Solvers for Verification of Neural Networks in Predictive Maintenance Applications

Interest in machine learning and neural networks has increased significantly in recent years. However, their applications are limited in safety-critical domains due to the lack of formal guarantees on their reliability and behavior. This paper shows recent advances in satisfiability modulo theory solvers used in the context of the verification of neural networks with piece-wise linear and transcendental activation functions. An experimental analysis is conducted using neural networks trained on a real-world predictive maintenance dataset. This study contributes to the research on enhancing the safety and reliability of neural networks through formal verification, enabling their deployment in safety-critical domains.

A Semidefinite Relaxation Based Branch-and-Bound Method for Tight Neural Network Verification

We introduce a novel method based on semidefinite program (SDP) for the tight and efficient verification of neural networks. The proposed SDP relaxation advances the present state of the art in SDP-based neural network verification by adding a set of linear constraints based on eigenvectors. We extend this novel SDP relaxation by combining it with a branch-and-bound method that can provably close the relaxation gap up to zero. We show formally that the proposed approach leads to a provably tighter solution than the present state of the art. We report experimental results showing that the proposed method outperforms baselines in terms of verified accuracy while retaining an acceptable computational overhead.

Iteratively Enhanced Semidefinite Relaxations for Efficient Neural Network Verification

We propose an enhanced semidefinite program (SDP) relaxation to enable the tight and efficient verification of neural networks (NNs). The tightness improvement is achieved by introducing a nonlinear constraint to existing SDP relaxations previously proposed for NN verification. The efficiency of the proposal stems from the iterative nature of the proposed algorithm in that it solves the resulting non-convex SDP by recursively solving auxiliary convex layer-based SDP problems. We show formally that the solution generated by our algorithm is tighter than state-of-the-art SDP-based solutions for the problem. We also show that the solution sequence converges to the optimal solution of the non-convex enhanced SDP relaxation. The experimental results on standard benchmarks in the area show that our algorithm achieves the state-of-the-art performance whilst maintaining an acceptable computational cost.

Incremental Verification of Neural Networks

Complete verification of deep neural networks (DNNs) can exactly determine whether the DNN satisfies a desired trustworthy property (e.g., robustness, fairness) on an infinite set of inputs or not. Despite the tremendous progress to improve the scalability of complete verifiers over the years on individual DNNs, they are inherently inefficient when a deployed DNN is updated to improve its inference speed or accuracy. The inefficiency is because the expensive verifier needs to be run from scratch on the updated DNN. To improve efficiency, we propose a new, general framework for incremental and complete DNN verification based on the design of novel theory, data structure, and algorithms. Our contributions implemented in a tool named IVAN yield an overall geometric mean speedup of 2.4x for verifying challenging MNIST and CIFAR10 classifiers and a geometric mean speedup of 3.8x for the ACAS-XU classifiers over the state-of-the-art baselines.

Towards rigorous understanding of neural networks via semantics-preserving transformations

AbstractIn this paper, we present an algebraic approach to the precise and global verification and explanation of Rectifier Neural Networks, a subclass of Piece-wise Linear Neural Networks (PLNNs), i.e., networks that semantically represent piece-wise affine functions. Key to our approach is the symbolic execution of these networks that allows the construction of semantically equivalent Typed Affine Decision Structures (TADS). Due to their deterministic and sequential nature, TADS can, similarly to decision trees, be considered as white-box models and therefore as precise solutions to the model and outcome explanation problem. TADS are linear algebras, which allows one to elegantly compare Rectifier Networks for equivalence or similarity, both with precise diagnostic information in case of failure, and to characterize their classification potential by precisely characterizing the set of inputs that are specifically classified, or the set of inputs where two network-based classifiers differ. All phenomena are illustrated along a detailed discussion of a minimal, illustrative example: the continuous XOR function.

Deep learning segmentation to analyze bubble dynamics and heat transfer during boiling at various pressures

Today, neural networks have increasingly gained the attention of researchers and become an effective instrument for a wide range of scientific applications, including issues related to the boiling. However, there are no universal tools in the literature that would allow detecting the life cycle of individual vapor bubbles and automatically measure a wide range of main boiling characteristics based on the high-speed visualization data. In this study, the U-net and Mask R-CNN convolutional neural networks were used to detect and segment bubbles obtained by visualization from the bottom side of a transparent heater during water boiling at various subatmospheric pressures. The key feature of the trained CNN architectures is the ability to detect bubbles located on a heated wall, while ignoring the bubbles that lift-off, and to determine the moment of their departure. The verification of various neural networks demonstrated that the Mask R-CNN architecture is more preferable to measure dynamic boiling characteristics. Through trained convolutional neural networks, a wide array of data on local boiling characteristics, including the nucleation site density, bubbles growth rate, life-time and departure diameters, waiting time between moments of bubbles departure and nucleation frequencies were automatically obtained for water boiling at various heat fluxes and pressures in the range of 42–103 kPa. Based on the parameters obtained, heat transfer simulation was carried out using various heat flux partitioning approaches and the ranges of their applicability were demonstrated.

Verification of Fuzzy Decision Trees

In recent years, there have been major strides in the safety verification of machine learning models such as neural networks and tree ensembles. However, fuzzy decision trees (FDT), also called soft or differentiable decision trees, are yet unstudied in the context of verification. They present unique verification challenges resulting from multiplications of input values; in the simplest case with a piecewise-linear splitting function, an FDT is piecewise-polynomial with degree up to the depth of the tree. We propose an abstraction-refinement algorithm for verification of properties of FDTs. We show that the problem is NP-Complete, like many other machine learning verification problems, and that our algorithm is complete in a finite precision setting. We benchmark on a selection of public data sets against an off-the-shelf SMT solver and a baseline variation of our algorithm that uses a refinement strategy from similar methods for neural network verification, finding the proposed method to be the fastest. Code for our algorithm along with our experiments and demos are available on GitHub at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/autonlab/fdt_verification</uri> .

The power of typed affine decision structures: a case study

AbstractTADS are a novel, concise white-box representation of neural networks. In this paper, we apply TADS to the problem of neural network verification, using them to generate either proofs or concise error characterizations for desirable neural network properties. In a case study, we consider the robustness of neural networks to adversarial attacks, i.e., small changes to an input that drastically change a neural networks perception, and show that TADS can be used to provide precise diagnostics on how and where robustness errors a occur. We achieve these results by introducing Precondition Projection, a technique that yields a TADS describing network behavior precisely on a given subset of its input space, and combining it with PCA, a traditional, well-understood dimensionality reduction technique. We show that PCA is easily compatible with TADS. All analyses can be implemented in a straightforward fashion using the rich algebraic properties of TADS, demonstrating the utility of the TADS framework for neural network explainability and verification. While TADS do not yet scale as efficiently as state-of-the-art neural network verifiers, we show that, using PCA-based simplifications, they can still scale to medium-sized problems and yield concise explanations for potential errors that can be used for other purposes such as debugging a network or generating new training samples.

Novel intelligent computing knacks for fuzzy dynamical model of vibrating mass system

This study is designed to analyze the fuzzy dynamical model of vibrating mass system by using neural networks (NNs). Stochastic numerical solvers are implemented with Levenberg–Marquardt backpropagation (LMB) algorithm. The proposed solvers handle the uncertainties of fuzzy differential system and minimize the error consistently. Validation, constancy and verification of NNs with LMB algorithm are endorsed by attaining reasonable accuracy through statistical measures in terms of error histograms, regression and mean squared errors for different values of fuzzy parameters.

Theoretical analysis of norm selection for robustness verification of neural networks

In the robustness verification of neural network, Lp-norm is generally used to measure the distance between two different data. However, different kinds of Lp-norm examine the specific differences between each dimension of the data. Some Lp-norm computational methods magnify specific differences between datasets. To verify the robustness of the network, the usual method is to calculate the different robustness radii of the L∞-norm. But which Lp norm performs best under a given network is rarely considered. To better reflect the nature of the network, this paper proposes a theoretical analysis of the robustness of the network that can be verified by choosing any norm type, and the relationship between the size of Lp-ball corresponding to different robust radii is proved. We conducted experiments on different classification networks and obtained a general norm selection method. As a result, a specific norm can be considered first for network robustness without computing all the different Lp-norm.

Abstract Layer for LeakyReLU for Neural Network Verification Based on Abstract Interpretation

Deep neural networks have been widely used in several complex tasks such as robotics, self-driving cars, medicine, etc. However, they have recently shown to be vulnerable in uncertain environments where inputs are noisy. As a consequence, the robustness of neural networks has become an essential property for their application in critical systems. Robustness is the capacity to take the same decision even when inputs are disturbed under different types of perturbations, including adversarial attacks. The great difficulty today is providing a formal guarantee of robustness, which is the context of this paper. To do so, abstract interpretation, a popular state-of-the-art method, consisting of converting the layers of the neural network into abstract layers, has been recently proposed. An abstract layer can act on a geometric abstract object or shape comprising implicitly an infinite number of inputs rather than an individual input. In this paper, we propose a new mathematical formulation of an abstract transformer to convert a LeakyReLU activation layer to an abstract layer. Moreover, we implement and integrate our transformer into the ERAN tool. For validation, we assess the performance of our transformer according to the LeakyReLU hyperparameter, and we study the robustness of the neural network according to the input perturbation intensity. Our approach is evaluated on three different datasets: MNIST, Fashion and a robotic dataset. The obtained results demonstrate the efficacy of our abstract transformer in terms of mathematical formulation and implementation.

Efficient Complete Verification of Neural Networks via Layerwised Splitting and Refinement

Safety and robustness properties are highly required for neural networks deployed in safety-critical applications. Current complete verification techniques of these properties suffer from the lack of efficiency and effectiveness. In this article, we present an efficient complete approach to verify safety and robustness properties of neural networks through incrementally determinizing activation states of neurons. The key idea is to generate constraints via layerwised splitting that make activation states of hidden neurons become deterministic efficiently. These constraints are then utilized for refining inputs systematically so that abstract analysis over the refined input can be more precise. Our approach decomposes a verification problem into a set of subproblems via layerwised input space splitting. The property is then checked on each subproblem, where the activation states of at least one hidden neurons will be determinized. Further checking is accelerated by constraint-guided input refinement. We have implemented a parallel tool called LayerSAR to verify safety and robustness properties of ReLU neural networks in a sound and complete way, and evaluated it extensively on several benchmark sets. Experimental results show that our approach is promising, compared with complete tools, such as Planet, Neurify, Marabou, ERAN, Venus, Venus2, and nnenum in verifying safety and robustness properties on the benchmarks.

Evaluation of Neural Network Verification Methods for Air-to-Air Collision Avoidance

Neural network approximations have become attractive to compress data for automation and autonomy algorithms for use on storage-limited and processing-limited aerospace hardware. However, unless these neural network approximations can be exhaustively verified to be safe, they cannot be certified for use on aircraft. An example of such systems is the unmanned Airborne Collision Avoidance System (ACAS) Xu, which is a very popular benchmark for open-loop neural network control system verification tools. This paper proposes a new closed-loop extension of this benchmark, which consists of a set of 10 closed-loop properties selected to evaluate the safety of an ownship aircraft in the presence of a co-altitude intruder aircraft. These closed-loop safety properties are used to evaluate five of the 45 neural networks that comprise the ACAS Xu benchmark (corresponding to co-altitude cases) as well as the switching logic between the five neural networks. The combination of nonlinear dynamics and switching between five neural networks is a challenging verification task accomplished with star-set reachability methods in two verification tools. The safety of the ownship aircraft under initial position uncertainty is guaranteed in every scenario proposed.