Related papers: Non-Linear Interactions in Neural Network Operator…

Proximal operators are now ubiquitous in non-smooth optimization. Since their introduction in the seminal work of Moreau, many papers have shown their effectiveness on a wide variety of problems, culminating in their use to construct…

We introduce Steerable Transformers, an extension of the Vision Transformer mechanism that maintains equivariance to the special Euclidean group $\mathrm{SE}(d)$. We propose an equivariant attention mechanism that operates on features…

Most statistical models for networks focus on pairwise interactions between nodes. However, many real-world networks involve higher-order interactions among multiple nodes, such as co-authors collaborating on a paper. Hypergraphs provide a…

Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of…

Koopman analysis provides a general framework from which to analyze a nonlinear dynamical system in terms of a linear operator acting on an infinite-dimensional observable space. This theoretical framework provides a rigorous underpinning…

Transformer architectures are typically described in algorithmic and statistical terms, leaving their internal mechanics without a familiar structural language for researchers trained in physical theories. To bridge this gap, we develop a…

Incorporating spatial information into hyperspectral unmixing procedures has been shown to have positive effects, due to the inherent spatial-spectral duality in hyperspectral scenes. Current research works that consider spatial information…

The analysis of symmetries is extremely useful across science. In Physics, symmetries are used to derive conservation laws and selection rules for transitions in interacting systems. In the early days of nonlinear optics (NLO), symmetries…

The most general operator product expansion in conformal field theory is obtained using the embedding space formalism and a new uplift for general quasi-primary operators. The uplift introduced here, based on quasi-primary operators with…

Training reinforcement learning (RL) agents often requires significant computational resources and prolonged training durations. To address this challenge, we build upon prior work that introduced a neural architecture with…

Behavior of neural networks is irremediably determined by the specific loss and data used during training. However it is often desirable to tune the model at inference time based on external factors such as preferences of the user or…

We study the derivative-informed learning of nonlinear operators between infinite-dimensional separable Hilbert spaces by neural networks. Such operators can arise from the solution of partial differential equations (PDEs), and are used in…

We investigate the optimization of neural networks on symmetric data, and compare the strategy of constraining the architecture to be equivariant to that of using data augmentation. Our analysis reveals that that the relative geometry of…

We introduce a Nemytskii neural operator framework for nonlinear model reduction of parametrized steady-state partial differential equations. The method generalizes reduced basis approaches by replacing linear combinations of basis…

Despite their central role in the success of foundational models and large-scale language modeling, the theoretical foundations governing the operation of Transformers remain only partially understood. Contemporary research has largely…

We establish in this work approximation results of deep neural networks for smooth functions measured in Sobolev norms, motivated by recent development of numerical solvers for partial differential equations using deep neural networks. {Our…

The subject of PT-symmetry and its areas of application have been blossoming over the past decade. Here, we consider a nonlinear Schr\"odinger model with a complex potential that can be tuned controllably away from being PT-symmetric, as it…

In machine learning, data is usually represented in a (flat) Euclidean space where distances between points are along straight lines. Researchers have recently considered more exotic (non-Euclidean) Riemannian manifolds such as hyperbolic…

Singularly perturbed problems present inherent difficulty due to the presence of a thin boundary layer in its solution. To overcome this difficulty, we propose using deep operator networks (DeepONets), a method previously shown to be…

Linear spaces with an Euclidean metric are ubiquitous in mathematics, arising both from quadratic forms and inner products. Operators on such spaces also occur naturally. In recent years, the study of multivariate operator theory has made…