Related papers: Deep Nonparametric Estimation of Operators between…
Implicit neural networks, a.k.a., deep equilibrium networks, are a class of implicit-depth learning models where function evaluation is performed by solving a fixed point equation. They generalize classic feedforward models and are…
We study the learnability of a class of compact operators known as Schatten--von Neumann operators. These operators between infinite-dimensional function spaces play a central role in a variety of applications in learning theory and inverse…
Neural operators (NOs) employ deep neural networks to learn mappings between infinite-dimensional function spaces. Deep operator network (DeepONet), a popular NO architecture, has demonstrated success in the real-time prediction of complex…
Next generation deep neural networks for classification hosted on embedded platforms will rely on fast, efficient, and accurate learning algorithms. Initialization of weights in learning networks has a great impact on the classification…
Deep Neural Networks (DNNs) excel at many tasks, often rivaling or surpassing human performance. Yet their internal processes remain elusive, frequently described as "black boxes." While performance can be refined experimentally, achieving…
Establishing a theoretical analysis that explains why deep learning can outperform shallow learning such as kernel methods is one of the biggest issues in the deep learning literature. Towards answering this question, we evaluate excess…
This paper presents a framework for bounding the approximation error in imitation model predictive controllers utilizing neural networks. Leveraging the Lipschitz properties of these neural networks, we derive a bound that guides dataset…
Deep neural networks have achieved tremendous success due to their representation power and adaptation to low-dimensional structures. Their potential for estimating structured regression functions has been recently established in the…
Operator learning is a recent development in the simulation of Partial Differential Equations (PDEs) by means of neural networks. The idea behind this approach is to learn the behavior of an operator, such that the resulting neural network…
Deep neural networks (DNNs) achieve remarkable predictive performance but remain difficult to interpret, largely due to overparameterization that obscures the minimal structure required for interpretation. Here we introduce DeepIn, a…
Optimal experimental design is a well studied field in applied science and engineering. Techniques for estimating such a design are commonly used within the framework of parameter estimation. Nonetheless, in recent years parameter…
This paper introduces \textbf{Measure Learning}, a paradigm for modeling ambiguity via non-linear expectations. We define Neural Expectation Operators as solutions to Backward Stochastic Differential Equations (BSDEs) whose drivers are…
Classical assumptions like strong convexity and Lipschitz smoothness often fail to capture the nature of deep learning optimization problems, which are typically non-convex and non-smooth, making traditional analyses less applicable. This…
We propose a very general framework for deriving rigorous bounds on the approximation error for physics-informed neural networks (PINNs) and operator learning architectures such as DeepONets and FNOs as well as for physics-informed operator…
Neural Operators that directly learn mappings between function spaces, such as Deep Operator Networks (DONs) and Fourier Neural Operators (FNOs), have received considerable attention. Despite the universal approximation guarantees for DONs…
One of the arguments to explain the success of deep learning is the powerful approximation capacity of deep neural networks. Such capacity is generally accompanied by the explosive growth of the number of parameters, which, in turn, leads…
We consider the problem of learning an unknown, possibly nonlinear operator between separable Hilbert spaces from supervised data. Inputs are drawn from a prescribed probability measure on the input space, and outputs are (possibly noisy)…
This study investigates leveraging stochastic gradient descent (SGD) to learn operators between general Hilbert spaces. We propose weak and strong regularity conditions for the target operator to depict its intrinsic structure and…
The skip-connections used in residual networks have become a standard architecture choice in deep learning due to the increased training stability and generalization performance with this architecture, although there has been limited…
We present a generalized version of the discretization-invariant neural operator and prove that the network is a universal approximation in the operator sense. Moreover, by incorporating additional terms in the architecture, we establish a…