Related papers: Generative Neural Operators of Log-Complexity Can …

Learn to Optimize (L2O) trains deep neural network-based solvers for optimization, achieving success in accelerating convex problems and improving non-convex solutions. However, L2O lacks rigorous theoretical backing for its own training…

Neural operators have become increasingly popular in solving \textit{partial differential equations} (PDEs) due to their superior capability to capture intricate mappings between function spaces over complex domains. However, the…

Continuous-depth neural networks, such as Neural ODEs, have refashioned the understanding of residual neural networks in terms of non-linear vector-valued optimal control problems. The common solution is to use the adjoint sensitivity…

Many problems in nonlinear analysis and optimization, among them variational inequalities and minimization of convex functions, can be reduced to finding zeros (namely, roots) of set-valued operators. Hence numerous algorithms have been…

We study the approximation rates of a class of deep neural network approximations of operators which arise as data-to-solution maps $\mathcal{S}$ of linear elliptic partial differential equations (PDEs), and act between pairs $X,Y$ of…

Deep neural networks are an attractive alternative for simulating complex dynamical systems, as in comparison to traditional scientific computing methods, they offer reduced computational costs during inference and can be trained directly…

While traditional Deep Learning (DL) optimization methods treat all training samples equally, Distributionally Robust Optimization (DRO) adaptively assigns importance weights to different samples. However, a significant gap exists between…

Neural operators extend data-driven models to map between infinite-dimensional functional spaces. While these operators perform effectively in either the time or frequency domain, their performance may be limited when applied to…

For partial differential equations on domains of arbitrary shapes, existing works of neural operators attempt to learn a mapping from geometries to solutions. It often requires a large dataset of geometry-solution pairs in order to obtain a…

Training deep neural networks for solving machine learning problems is one great challenge in the field, mainly due to its associated optimisation problem being highly non-convex. Recent developments have suggested that many training…

Current neural operators often struggle to generalize to complex, out-of-distribution conditions, limiting their ability in seismic wavefield representation. To address this, we propose a generative neural operator (GNO) that leverages…

Graduated optimization is a global optimization technique that is used to minimize a multimodal nonconvex function by smoothing the objective function with noise and gradually refining the solution. This paper experimentally evaluates the…

Optimization problems are ubiquitous in our societies and are present in almost every segment of the economy. Most of these optimization problems are NP-hard and computationally demanding, often requiring approximate solutions for…

Despite their remarkable success in approximating a wide range of operators defined by PDEs, existing neural operators (NOs) do not necessarily perform well for all physics problems. We focus here on high-frequency waves to highlight…

The neural operator (NO) framework has emerged as a powerful tool for solving partial differential equations (PDEs). Recent NOs are dominated by the Transformer architecture, which offers NOs the capability to capture long-range…

In this article, we propose a topological model to encode partial equivariance in neural networks. To this end, we introduce a class of operators, called P-GENEOs, that change data expressed by measurements, respecting the action of certain…

We consider neural networks with a single hidden layer and non-decreasing homogeneous activa-tion functions like the rectified linear units. By letting the number of hidden units grow unbounded and using classical non-Euclidean…

Neural operator architectures approximate operators between infinite-dimensional Banach spaces of functions. They are gaining increased attention in computational science and engineering, due to their potential both to accelerate…

Neural operators (NOs) provide a new paradigm for efficiently solving partial differential equations (PDEs), but their training depends on costly high-fidelity data from numerical solvers, limiting applications in complex systems. We…

Neural surrogate solvers can estimate solutions to partial differential equations in physical problems more efficiently than standard numerical methods, but require extensive high-resolution training data. In this paper, we break this…