11 papers
Multilevel Picard approximations for McKean-Vlasov stochastic differential equations with nonconstant diffusion
Ariel Neufeld, Tuan Anh Nguyen, Philipp Schmocker
We introduce multilevel Picard (MLP) approximations for McKean--Vlasov stochastic differential equations (SDEs) with nonconstant diffusion coefficient. Under standard Lipschitz ass…
Universal approximation property of Banach space-valued random feature models including random neural networks
Ariel Neufeld, Philipp Schmocker
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 initi…
Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in -sense
Ariel Neufeld, Tuan Anh Nguyen
We prove that multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation are capable of approximating solutions of semilinear Kolmogor…
Solving stochastic partial differential equations using neural networks in the Wiener chaos expansion
Ariel Neufeld, Philipp Schmocker
In this paper, we solve stochastic partial differential equations (SPDEs) numerically by using (possibly random) neural networks in the truncated Wiener chaos expansion of their co…
Approximation rates of quantum neural networks for periodic functions via Jackson's inequality
Ariel Neufeld, Philipp Schmocker, Viet Khoa Tran
Quantum neural networks (QNNs) are an analog of classical neural networks in the world of quantum computing, which are represented by a unitary matrix with trainable parameters. In…
Generative Neural Operators of Log-Complexity Can Simultaneously Solve Infinitely Many Convex Programs
Anastasis Kratsios, Ariel Neufeld, Philipp Schmocker
Neural operators (NOs) are a class of deep learning models designed to simultaneously solve infinitely many related problems by casting them into an infinite-dimensional space, whe…