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The recently introduced DeepONet operator-learning framework for PDE control is extended from the results for basic hyperbolic and parabolic PDEs to an advanced hyperbolic class that involves delays on both the state and the system output…
Accurately quantifying long-term risk probabilities in diverse stochastic systems is essential for safety-critical control. However, existing sampling-based and partial differential equation (PDE)-based methods often struggle to handle…
Operator learning is a variant of machine learning that is designed to approximate maps between function spaces from data. The Fourier Neural Operator (FNO) is one of the main model architectures used for operator learning. The FNO combines…
We present an end-to-end framework to learn partial differential equations that brings together initial data production, selection of boundary conditions, and the use of physics-informed neural operators to solve partial differential…
This focused review explores a range of neural operator architectures for approximating solutions to parametric partial differential equations (PDEs), emphasizing high-level concepts and practical implementation strategies. The study covers…
For the class of de Branges-Rovnyak spaces $\mathcal{H}(b)$ of the unit disk $\mathbb{D}$ defined by extreme points $b$ of the unit ball of $H^\infty$, we study the problem of approximation of a general function in $\mathcal{H}(b)$ by a…
Neural operators generalize neural networks to learn mappings between function spaces from data. They are commonly used to learn solution operators of parametric partial differential equations (PDEs) or propagators of time-dependent PDEs.…
We approximate functions defined on smooth bounded domains by elements of the eigenspaces of the Laplacian or the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces. We prove…
Recently, Implicit Neural Representations (INRs) parameterized by neural networks have emerged as a powerful and promising tool to represent different kinds of signals due to its continuous, differentiable properties, showing superiorities…
Smooth activation functions are ubiquitous in modern deep learning, yet their theoretical advantages over non-smooth counterparts remain poorly understood. In this work, we study both approximation and statistical properties of neural…
The finite element method is an indispensable tool in engineering, but its computational complexity prevents applications for control or at system-level. Model order reduction bridges this gap, creating highly efficient yet accurate…
We establish upper bounds for the weak and strong error resulting from a perturbation of the noise driving the stochastic Burgers equation, where we assume the noise to be additive and of trace class and the initial value to be sufficiently…
Neural operators have emerged as fast surrogate models for physics simulations, yet they remain acutely vulnerable to adversarial perturbations, a critical liability for safety-critical digital twin deployments. We present a synergistic…
Neural operator learning directly constructs the mapping relationship from the equation parameter space to the solution space, enabling efficient direct inference in practical applications without the need for repeated solution of partial…
While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear…
We propose a family of quantum algorithms for estimating Gowers uniformity norms $ U^k $ over finite abelian groups and demonstrate their applications to testing polynomial structure and counting arithmetic progressions. Building on recent…
Deep learning surrogate models have shown promise in solving partial differential equations (PDEs). Among them, the Fourier neural operator (FNO) achieves good accuracy, and is significantly faster compared to numerical solvers, on a…
Motivated by the rapidly growing field of mathematics for operator approximation with neural networks, we present a novel universal operator approximation theorem for a broad class of encoder-decoder architectures. In this study, we focus…
We propose the Factorized Fourier Neural Operator (F-FNO), a learning-based approach for simulating partial differential equations (PDEs). Starting from a recently proposed Fourier representation of flow fields, the F-FNO bridges the…
A burgeoning line of research leverages deep neural networks to approximate the solutions to high dimensional PDEs, opening lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of…