相关论文: Universal Approximation of Nonlinear Operators and…
Neural networks have the ability to serve as universal function approximators, but they are not interpretable and don't generalize well outside of their training region. Both of these issues are problematic when trying to apply standard…
Neural operators have emerged as transformative tools for learning mappings between infinite-dimensional function spaces, offering useful applications in solving complex partial differential equations (PDEs). This paper presents a rigorous…
An important application of neural networks to scientific computing has been the learning of non-linear operators. In this framework, a neural network is trained to fit a non-linear map between two infinite dimensional spaces, for example,…
Universal approximation theorems are the foundations of classical neural networks, providing theoretical guarantees that the latter are able to approximate maps of interest. Recent results have shown that this can also be achieved in a…
We develop an $\mathrm{L}^p$-theory for Dolbeault-Dirac operators on compact K\"ahler manifolds with coefficients in a Hermitian holomorphic vector bundle $E$. For each $p \in (1,\infty)$ we consider the closed $\mathrm{L}^p$-realization…
Complex dynamic systems are typically either modeled using expert knowledge in the form of differential equations or via data-driven universal approximation models such as artificial neural networks (ANN). While the first approach has…
Neural operator learning models have emerged as very effective surrogates in data-driven methods for partial differential equations (PDEs) across different applications from computational science and engineering. Such operator learning…
Learning operators mapping between infinite-dimensional Banach spaces via neural networks has attracted a considerable amount of attention in recent years. In this paper, we propose an interfaced operator network (IONet) to solve parametric…
In this paper four iterative algorithms for learning analysis operators are presented. They are built upon the same optimisation principle underlying both Analysis K-SVD and Analysis SimCO. The Forward and Sequential Analysis Operator…
Neural ordinary differential equations (neural ODEs) are a popular type of deep learning model that operate with continuous-depth architectures. To assess how well such models perform on unseen data, it is crucial to understand their…
In the framework of operator theory, we investigate a close Lie theoretic relationship between all operator ideals and certain classical groups of invertible operators that can be described as the solution sets of certain algebraic…
Quantum speed-ups for dynamical simulation usually demand unitary time-evolution, whereas the large ODE/PDE systems encountered in realistic physical models are generically non-unitary. We present a universal moment-fulfilling dilation that…
Classical neural networks are known for their ability to approximate mappings between finite-dimensional spaces, but they fall short in capturing complex operator dynamics across infinite-dimensional function spaces. Neural operators, in…
Neural networks (NNs) are known for their high predictive accuracy in complex learning problems. Beside practical advantages, NNs also indicate favourable theoretical properties such as universal approximation (UA) theorems. Binarized…
A key challenge in scientific machine learning is solving partial differential equations (PDEs) on complex domains, where the curved geometry complicates the approximation of functions and their derivatives required by differential…
Although very successfully used in conventional machine learning, convolution based neural network architectures -- believed to be inconsistent in function space -- have been largely ignored in the context of learning solution operators of…
We establish fundamental mathematical limits on universal approximation theorem (UAT) system alignment by proving that catastrophic failures are an inescapable feature of any useful computational system. Our central thesis is that for any…
Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal semi-finite trace $\tau$ and let $S_0(\tau)$ be the algebra of all $\tau$-compact operators affiliated with $\mathcal{M}$. Let $E(\tau)\subseteq S_0(\tau)$ be a…
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…
We introduce a Banach space-valued extension of random feature learning, a data-driven supervised machine learning technique for large-scale kernel approximation. By randomly initializing the feature maps, only the linear readout needs to…