Universal approximation serves as the foundation of neural network learning algorithms. However, some networks establish their universal approximation property by demonstrating that the iterative errors converge in probability measure rather than the more rigorous norm convergence, which makes the universal approximation property of randomized learning networks highly sensitive to random parameter selection, Broad residual learning system (BRLS), as a member of randomized learning models, also encounters this issue. We theoretically demonstrate the limitation of its universal approximation property, that is, the iterative errors do not satisfy norm convergence if the selection of random parameters is inappropriate and the convergence rate meets certain conditions. To address this issue, we propose the broad stochastic configuration residual learning system (BSCRLS) algorithm, which features a novel supervisory mechanism adaptively constraining the range settings of random parameters on the basis of BRLS framework, Furthermore, we prove the universal approximation theorem of BSCRLS based on the more stringent norm convergence. Three versions of incremental BSCRLS algorithms are presented to satisfy the application requirements of various network updates. Solar panels dust detection experiments are performed on publicly available dataset and compared with 13 deep and broad learning algorithms. Experimental results reveal the effectiveness and superiority of BSCRLS algorithms.

Invertible neural networks based on coupling flows (CF-INNs) have various machine learning applications such as image synthesis and representation learning.

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 (e.g., fully connected networks, CNNs, or Transformers) and then prove their universal approximability. However, this approach suffers from two major limitations: first, every newly proposed architecture often requires a completely new proof from scratch; second, these proofs are largely isolated from one another, lacking a common analytical foundation. This not only incurs significant redundancy but also hinders unified theoretical understanding across different network families. To address these issues, this paper proposes a general and modular framework for proving universal approximation. We define a basic building block (comprising one or multiple layers) that possesses the universal approximation property as a Universal Approximation Module (UAM). Under this condition, we show that any deep network composed of such modules inherently retains the universal approximation property. Moreover, the overall approximation process can be interpreted as a progressive refinement across modules. This perspective not only unifies the analysis of diverse architectures but also enables a step-by-step understanding of how expressive power evolves through the network.

1-Lipschitz neural networks are fundamental for generative modelling, inverse problems, and robust classifiers. In this paper, we focus on 1-Lipschitz residual networks (ResNets) based on explicit Euler steps of negative gradient flows and study their approximation capabilities. Leveraging the Restricted Stone-Weierstrass Theorem, we first show that these 1-Lipschitz ResNets are dense in the set of scalar 1-Lipschitz functions on any compact domain when width and depth are allowed to grow. We also show that these networks can exactly represent scalar piecewise affine 1-Lipschitz functions. We then prove a stronger statement: by inserting norm-constrained linear maps between the residual blocks, the same density holds when the hidden width is fixed. Because every layer obeys simple norm constraints, the resulting models can be trained with off-the-shelf optimisers. This paper provides the first universal approximation guarantees for 1-Lipschitz ResNets, laying a rigorous foundation for their practical use.

As discussed in section 4.4.

Neural Ordinary Differential Equations (Neural ODEs), which are the continuous-time analog of Residual Neural Networks (ResNets), have gained significant attention in recent years. Similarly, Neural Delay Differential Equations (Neural DDEs) can be interpreted as an infinite depth limit of Densely Connected Residual Neural Networks (DenseResNets). In contrast to traditional ResNet architectures, DenseResNets are feed-forward networks that allow for shortcut connections across all layers. These additional connections introduce memory in the network architecture, as typical in many modern architectures. In this work, we explore how the memory capacity in neural DDEs influences the universal approximation property. The key parameter for studying the memory capacity is the product $K τ$ of the Lipschitz constant and the delay of the DDE. In the case of non-augmented architectures, where the network width is not larger than the input and output dimensions, neural ODEs and classical feed-forward neural networks cannot have the universal approximation property. We show that if the memory capacity $Kτ$ is sufficiently small, the dynamics of the neural DDE can be approximated by a neural ODE. Consequently, non-augmented neural DDEs with a small memory capacity also lack the universal approximation property. In contrast, if the memory capacity $Kτ$ is sufficiently large, we can establish the universal approximation property of neural DDEs for continuous functions. If the neural DDE architecture is augmented, we can expand the parameter regions in which universal approximation is possible. Overall, our results show that by increasing the memory capacity $Kτ$, the infinite-dimensional phase space of DDEs with positive delay $τ>0$ is not sufficient to guarantee a direct jump transition to universal approximation, but only after a certain memory threshold, universal approximation holds.

We identify various classes of neural networks that are able to approximate continuous functions locally uniformly subject to fixed global linear growth constraints. For such neural networks the associated neural stochastic differential equations can approximate general stochastic differential equations, both of It\^o diffusion type, arbitrarily well. Moreover, quantitative error estimates are derived for stochastic differential equations with sufficiently regular coefficients.

We study the approximation properties of shallow neural networks whose activation function is defined as the flow of a neural ordinary differential equation (neural ODE) at the final time of the integration interval. We prove the universal approximation property (UAP) of such shallow neural networks in the space of continuous functions. Furthermore, we investigate the approximation properties of shallow neural networks whose parameters are required to satisfy some constraints. In particular, we constrain the Lipschitz constant of the flow of the neural ODE to increase the stability of the shallow neural network, and we restrict the norm of the weight matrices of the linear layers to one to make sure that the restricted expansivity of the flow is not compensated by the increased expansivity of the linear layers. For this setting, we prove approximation bounds that tell us the accuracy to which we can approximate a continuous function with a shallow neural network with such constraints. We prove that the UAP holds if we consider only the constraint on the Lipschitz constant of the flow or the unit norm constraint on the weight matrices of the linear layers.