Related papers: Operator Learning of Lipschitz Operators: An Infor…
We investigate the effect of explicitly enforcing the Lipschitz continuity of neural networks with respect to their inputs. To this end, we provide a simple technique for computing an upper bound to the Lipschitz constant---for multiple…
A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically…
This paper is concerned with the training of neural networks (NNs) under semidefinite constraints, which allows for NN training with robustness and stability guarantees. In particular, we focus on Lipschitz bounds for NNs. Exploiting the…
We present an algorithm for learning operators between Banach spaces, based on the use of Leray-Schauder mappings to learn a finite-dimensional approximation of compact subspaces. We show that the resulting method is a universal…
This paper is concerned with fundamental limits on the approximation of nonlinear dynamical systems. Specifically, we show that recurrent neural networks (RNNs) can approximate nonlinear systems -- that satisfy a Lipschitz property and…
In this paper, we study {\it operator spaces\/} in the sense of the theory developed recently by Blecher-Paulsen [BP] and Effros-Ruan [ER1]. By an operator space, we mean a closed subspace $E\subset B(H)$, with $H$ Hilbert. We will be…
This paper studies the learning of linear operators between infinite-dimensional Hilbert spaces. The training data comprises pairs of random input vectors in a Hilbert space and their noisy images under an unknown self-adjoint linear…
On the one hand, the framework of mixed norm spaces has potential applications in different areas of mathematics. On the other hand, neural network (NN) operators are well established as approximators, attracting significant attention in…
Imaging systems are commonly described using resolution, contrast, and signal-to-noise ratio, but these quantities do not provide a general account of how physical transformations affect the flow of information. This paper introduces an…
Operator learning problems arise in many key areas of scientific computing where Partial Differential Equations (PDEs) are used to model physical systems. In such scenarios, the operators map between Banach or Hilbert spaces. In this work,…
Neural networks (NNs) have emerged as a state-of-the-art method for modeling nonlinear systems in model predictive control (MPC). However, the robustness of NNs, in terms of sensitivity to small input perturbations, remains a critical…
While many approaches exist in the literature to learn low-dimensional representations for data collections in multiple modalities, the generalizability of multi-modal nonlinear embeddings to previously unseen data is a rather overlooked…
The Lipschitz space of an infinite (locally-finite) graph is defined as the set of functions on the vertices of the graph such that the differences of the values between adjacent vertices remain bounded. In this paper we prove that this set…
We use the notion of $\A$-compact sets, which are determined by a Banach operator ideal $\A$, to show that most classic results of certain approximation properties and several Banach operator ideals can be systematically studied under this…
In Artificial Intelligence (AI) and computational science, learning the mappings between functions (called operators) defined on complex computational domains is a common theoretical challenge. Recently, Neural Operator emerged as a…
The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators,…
Transformers, and the attention mechanism in particular, have become ubiquitous in machine learning. Their success in modeling nonlocal, long-range correlations has led to their widespread adoption in natural language processing, computer…
At its core, machine learning seeks to train models that reliably generalize beyond noisy observations; however, the theoretical vacuum in which state-of-the-art universal approximation theorems (UATs) operate isolates them from this goal,…
Neural operators are neural network-based surrogate models for approximating solution operators of parametric partial differential equations, enabling efficient many-query computations in science and engineering. Many applications,…
Given a Banach lattice $L,$ the space of lattice Lipschitz operators on $L$ has been introduced as a natural Lipschitz generalization of the linear notions of diagonal operator and multiplication operator on Banach function lattices. It is…