arXiv:2406. 12264v5 Announce Type: replace-cross Abstract: We obtain a new universal approximation theorem for continuous (possibly nonlinear) operators on arbitrary Banach spaces using the Leray-Schauder mapping.
arXiv:2503. 24092v2 Announce Type: replace-cross Abstract: Motivated by the rapidly growing field of mathematics for operator approximation with neural networks, we present a novel universal operator approximation theorem for broad classes of encoder-decoder architectures and a wide range of input and output spaces.
arXiv:2403.12187v2 Announce Type: replace-cross Abstract: Motivated by the abundance of functional data, such as time series and images, we study the approximation and statistical learning of nonline...
arXiv:2608. 11831v1 Announce Type: new Abstract: Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning.
arXiv:2402.04691v5 Announce Type: replace-cross Abstract: This study investigates the use of stochastic gradient descent (SGD) to learn operators between general Hilbert spaces. We study weak and str...
arXiv:2507.07292v2 Announce Type: replace Abstract: We develop a new and general encode-approximate-reconstruct operator learning model that leverages learned neural representations of bases for inpu...
arXiv:2606. 01244v1 Announce Type: cross Abstract: We study operator learning using encoder--decoder neural networks.
arXiv:2503. 18219v2 Announce Type: replace Abstract: This work studies the sampling complexity of learning with ReLU neural networks and neural operators.
Polynomial-Augmented Neural Networks (PANNs) merge deep neural networks with polynomial expansions to leverage the flexibility of DNNs and the rapid convergence of polynomials. The architecture introduces orthogonality constraints, basis pruning, and polynomial preconditioning to stabilize training and improve accuracy across diverse problems. Experiments show that PANNs outperform both pure DNNs and polynomial methods in approximating smooth and limited‑smoothness functions, as well as in solving partial differential equations.
arXiv:2606. 17419v1 Announce Type: new Abstract: We develop approximation and generalization error estimates for multi-input neural operators, with the output error measured in Sobolev norms.
arXiv:2606. 09820v1 Announce Type: cross Abstract: We generalize the universal approximation theorem for functional input neural networks (FNN) to differentiable maps by including the approximation of the derivatives.
The paper investigates how many linear samples are needed to learn Lipschitz operators under Gaussian measures. It establishes both lower and upper bounds on the Hermite polynomial approximation error and shows that the minimal worst‑case error cannot converge algebraically with the number of samples. However, if the covariance operator of the Gaussian measure decays rapidly, convergence rates arbitrarily close to any algebraic rate can be achieved.
arXiv:2603. 00819v2 Announce Type: replace-cross Abstract: This paper surveys recent developments at the intersection of operator learning, statistical learning theory, and approximation theory.