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Formal verification of neural networks is critical for their safe adoption in real-world applications. However, designing a precise and scalable verifier which can handle different activation functions, realistic network architectures and…
Recently, formal verification of deep neural networks (DNNs) has garnered considerable attention, and over-approximation based methods have become popular due to their effectiveness and efficiency. However, these strategies face challenges…
In this paper we investigate formal verification problems for Neural Network computations. Various reachability problems will be in the focus, such as: Given symbolic specifications of allowed inputs and outputs in form of Linear…
Fully connected deep neural networks are successfully applied to classification and function approximation problems. By minimizing the cost function, i.e., finding the proper weights and biases, models can be built for accurate predictions.…
We derive an approximation error bound that holds simultaneously for a function and all its derivatives up to any prescribed order. The bounds apply to elementary functions, including multivariate polynomials, the exponential function, and…
We introduce an abstract neural flow framework for neural networks and neural operators. The framework contains two continuous-depth models, namely neural flows with composition and separation structures, and covers both finite-dimensional…
Within the limited scope of this paper, we argue that artificial general intelligence cannot emerge from current neural network paradigms regardless of scale, nor is such an approach healthy for the field at present. Drawing on various…
The possibility of approximating a continuous function on a compact subset of the real line by a feedforward single hidden layer neural network with a sigmoidal activation function has been studied in many papers. Such networks can…
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…
Interval arithmetic is a simple way to compute a mathematical expression to an arbitrary accuracy, widely used for verifying floating-point computations. Yet this simplicity belies challenges. Some inputs violate preconditions or cause…
We consider the Bayesian optimal filtering problem: i.e. estimating some conditional statistics of a latent time-series signal from an observation sequence. Classical approaches often rely on the use of assumed or estimated transition and…
Deep neural networks' remarkable ability to correctly fit training data when optimized by gradient-based algorithms is yet to be fully understood. Recent theoretical results explain the convergence for ReLU networks that are wider than…
Density functional theory is one of the most efficient and widely used computational methods of quantum mechanics, especially in fields such as solid state physics and quantum chemistry. From the theoretical perspecive, its central object…
In this article we present new results on neural networks with linear threshold activation functions. We precisely characterize the class of functions that are representable by such neural networks and show that 2 hidden layers are…
The success of Neural networks in providing miraculous results when applied to a wide variety of tasks is astonishing. Insight in the working can be obtained by studying the universal approximation property of neural networks. It is proved…
Neural networks are one of the most popularly used methods in machine learning and artificial intelligence nowadays. Due to the universal approximation theorem (Hornik et al. (1989)), a neural network with one hidden layer can approximate…
Universal approximation theorems establish the expressive capacity of neural network architectures. For dynamical systems, existing results are limited to finite time horizons or systems with a globally stable equilibrium, leaving…
Neural network based approximate computing is a universal architecture promising to gain tremendous energy-efficiency for many error resilient applications. To guarantee the approximation quality, existing works deploy two neural networks…
Neural ordinary differential equations (NODEs) is an invertible neural network architecture promising for its free-form Jacobian and the availability of a tractable Jacobian determinant estimator. Recently, the representation power of NODEs…
This work studies approximation based on single-hidden-layer feedforward and recurrent neural networks with randomly generated internal weights. These methods, in which only the last layer of weights and a few hyperparameters are optimized,…