Related papers: Universal Approximation Theorem and error bounds f…

We present a numerical scheme for computation of Artificial Neural Networks (ANN) weights, which stems from the Universal Approximation Theorem, avoiding laborious iterations. The proposed algorithm adheres to the underlying theory, is…

Fault-tolerant quantum computers offer the promise of dramatically improving machine learning through speed-ups in computation or improved model scalability. In the near-term, however, the benefits of quantum machine learning are not so…

This paper studies the approximation capacity of neural networks with an arbitrary activation function and with norm constraint on the weights. Upper and lower bounds on the approximation error of these networks are computed for smooth…

Introducing quantum sensors as solution to real-world problem demands reliability and controllability outside laboratory conditions. Producers and operators ought to be assumed to have limited resources ready available for calibration, and…

Approximation capability of reservoir systems whose reservoir is a recurrent neural network (RNN) is discussed. We show what we call uniform strong universality of RNN reservoir systems for a certain class of dynamical systems. This means…

The universal approximation theorem establishes that neural networks can approximate any continuous function on a compact set. Later works in approximation theory provide quantitative approximation rates for ReLU networks on the class of…

We design a quantum version of neural networks with sinusoidal activation functions and compare its performance to the classical case. We create a general quantum sine circuit implementing a discretised sinusoidal activation function. Along…

We study the approximation of multivariate functions with tensor networks (TNs), providing some answers to the following two questions: ``what are the approximation capabilities of TNs for functions from classical smoothness classes?'' and…

Universal approximation theory offers a foundational framework to verify neural network expressiveness, enabling principled utilization in real-world applications. However, most existing theoretical constructions are established by…

We propose a novel class of neural network-like parametrized functions, i.e., general transformation neural networks (GTNNs), for high-dimensional approximation. Conventional deep neural networks sometimes perform less accurately on…

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…

We investigate the approximation capabilities of dense neural networks. While universal approximation theorems establish that sufficiently large architectures can approximate arbitrary continuous functions if there are no restrictions on…

The universal approximation property is fundamental to the success of neural networks, and has traditionally been achieved by training networks without any constraints on their parameters. However, recent experimental research proposed a…

As demonstrated in many areas of real-life applications, neural networks have the capability of dealing with high dimensional data. In the fields of optimal control and dynamical systems, the same capability was studied and verified in many…

Quantum neural networks generalize classical artificial neural networks into the quantum domain. They are formulated as parameterized quantum circuits which are optimized by measuring and minimizing a suitably chosen loss function. The core…

Deep learning architectures are highly diverse. To prove their universal approximation properties, existing works typically rely on model-specific proofs. Generally, they construct a dedicated mathematical formulation for each architecture…

We present a simple proof of the approximate Eastin-Knill theorem, which connects the quality of a quantum error-correcting code (QECC) with its ability to achieve a universal set of transversal logical gates. Our derivation employs…

Landmark universal function approximation results for neural networks with trained weights and biases provided the impetus for the ubiquitous use of neural networks as learning models in neuroscience and Artificial Intelligence (AI). Recent…

In this work, we consider the approximation of a large class of bounded functions, with minimal regularity assumptions, by ReLU neural networks. We show that the approximation error can be bounded from above by a quantity proportional to…

Recently, deep Convolutional Neural Networks (CNNs) have proven to be successful when employed in areas such as reduced order modeling of parametrized PDEs. Despite their accuracy and efficiency, the approaches available in the literature…