Related papers: Topological DeepONets and a generalization of the …
Universal approximation theorems provide a mathematical explanation for the expressive power of neural networks. They assert that, under mild conditions on the activation function, feedforward neural networks are dense in broad function…
We define a neural network in infinite dimensional spaces for which we can show the universal approximation property. Indeed, we derive approximation results for continuous functions from a Fr\'echet space $\X$ into a Banach space $\Y$. The…
This paper develops fundamental limits of deep neural network learning by characterizing what is possible if no constraints are imposed on the learning algorithm and on the amount of training data. Concretely, we consider Kolmogorov-optimal…
In this paper, we propose a multiscale DeepONet to represent nonlinear operator between Banach spaces of highly oscillatory continuous functions. The multiscale deep neural network (DNN) utilizes a multiple scaling technique to convert high…
This paper develops a randomized approach for incrementally building deep neural networks, where a supervisory mechanism is proposed to constrain the random assignment of the weights and biases, and all the hidden layers have direct links…
Our work intends to show that: (1) Quantum Neural Networks (QNN) can be mapped onto spinnetworks, with the consequence that the level of analysis of their operation can be carried out on the side of Topological Quantum Field Theories…
Existing architectures for operator learning require that the number and locations of sensors (where the input functions are evaluated) remain the same across all training and test samples, significantly restricting the range of their…
A novel approach to approximate solutions of Stochastic Differential Equations (SDEs) by Deep Neural Networks is derived and analysed. The architecture is inspired by the notion of Deep Operator Networks (DeepONets), which is based on…
Deep operator networks (DeepONets) represent a powerful class of data-driven methods for operator learning, demonstrating strong approximation capabilities for a wide range of linear and nonlinear operators. They have shown promising…
We present a general kernel-based framework for learning operators between Banach spaces along with a priori error analysis and comprehensive numerical comparisons with popular neural net (NN) approaches such as Deep Operator Net (DeepONet)…
Parametric PDEs power modern simulation, design, and digital-twin systems, yet their many-query workloads still hinge on repeatedly solving large finite-element systems. Existing operator-learning approaches accelerate this process but…
Based on the tree architecture, the objective of this paper is to design deep neural networks with two or more hidden layers (called deep nets) for realization of radial functions so as to enable rotational invariance for near-optimal…
Motivated by the developing mathematics of deep learning, we build universal functions approximators of continuous maps between arbitrary Polish metric spaces $\mathcal{X}$ and $\mathcal{Y}$ using elementary functions between Euclidean…
We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider feedforward networks with a complex activation function $\sigma : \mathbb{C} \to…
We present the Input-Connected Multilayer Perceptron (IC-MLP), a feedforward neural network architecture in which each hidden neuron receives, in addition to the outputs of the preceding layer, a direct affine connection from the raw input.…
This work explores the application of deep operator learning principles to a problem in statistical physics. Specifically, we consider the linear kinetic equation, consisting of a differential advection operator and an integral collision…
Convolutional neural networks (CNN) are built upon the classical McCulloch-Pitts neuron model, which is essentially a linear model, where the nonlinearity is provided by a separate activation function. Several researchers have proposed…
Deep learning employs multi-layer neural networks trained via the backpropagation algorithm. This approach has achieved success across many domains and relies on adaptive gradient methods such as the Adam optimizer. Sequence modeling…
Accurately learning solution operators for time-dependent partial differential equations (PDEs) from sparse and irregular data remains a challenging task. Recurrent DeepONet extensions inherit the discrete-time limitations of…
Maxwell's equations, a system of linear partial differential equations (PDEs), describe the behavior of electric and magnetic fields in time and space and are essential for many important electromagnetic applications. Although numerical…