Related papers: Solving parametric elliptic interface problems via…

Scientific computing has been an indispensable tool in applied sciences and engineering, where traditional numerical methods are often employed due to their superior accuracy guarantees. However, these methods often encounter challenges…

Deep operator networks (DeepONets) are receiving increased attention thanks to their demonstrated capability to approximate nonlinear operators between infinite-dimensional Banach spaces. However, despite their remarkable early promise,…

Learning operators for parametric partial differential equations (PDEs) using neural networks has gained significant attention in recent years. However, standard approaches like Deep Operator Networks (DeepONets) require extensive labeled…

Singularly perturbed problems present inherent difficulty due to the presence of a thin boundary layer in its solution. To overcome this difficulty, we propose using deep operator networks (DeepONets), a method previously shown to be…

As an emerging paradigm in scientific machine learning, neural operators aim to learn operators, via neural networks, that map between infinite-dimensional function spaces. Several neural operators have been recently developed. However, all…

Partial differential equations (PDEs) underlie our understanding and prediction of natural phenomena across numerous fields, including physics, engineering, and finance. However, solving parametric PDEs is a complex task that necessitates…

In this paper, we present a framework for learning the solution map of a backward parabolic Cauchy problem. The solution depends continuously but nonlinearly on the final data, source, and force terms, all residing in Banach spaces of…

We introduce a novel neural operator architecture designed to approximate solutions of linear elliptic partial differential equations with high-contrast, spatially varying coefficients. The network, termed the Iterated V-shaped Net…

This article explores operator learning models that can deduce solutions to partial differential equations (PDEs) on arbitrary domains without requiring retraining. We introduce two innovative models rooted in boundary integral equations…

We present $\phi-$DeepONet, a physics-informed neural operator designed to learn mappings between function spaces that may contain discontinuities or exhibit non-smooth behavior. Classical neural operators are based on the universal…

The transferable neural network (TransNet) is a two-layer shallow neural network with pre-determined and uniformly distributed neurons in the hidden layer, and the least-squares solvers can be particularly used to compute the parameters of…

In this paper, we investigate the applications of operator learning, specifically DeepONet, for solving nonlinear partial differential equations (PDEs). Unlike conventional function learning methods that require training separate neural…

DeepONets have recently been proposed as a framework for learning nonlinear operators mapping between infinite dimensional Banach spaces. We analyze DeepONets and prove estimates on the resulting approximation and generalization errors. In…

Interface problems pose significant challenges due to the discontinuity of their solutions, particularly when they involve singular perturbations or high-contrast coefficients, resulting in intricate singularities that complicate…

We construct and analyze approximation rates of deep operator networks (ONets) between infinite-dimensional spaces that emulate with an exponential rate of convergence the coefficient-to-solution map of elliptic second-order partial…

In this paper, we evaluate the effectiveness of deep operator networks (DeepONets) in solving both forward and inverse problems of partial differential equations (PDEs) on unknown manifolds. By unknown manifolds, we identify the manifold by…

We introduce an output layer for neural networks that ensures satisfaction of convex constraints. Our approach, $\Pi$net, leverages operator splitting for rapid and reliable projections in the forward pass, and the implicit function theorem…

Solving partial differential equations (PDEs) by learning the solution operators has emerged as an attractive alternative to traditional numerical methods. However, implementing such architectures presents two main challenges: flexibility…

Operator regression provides a powerful means of constructing discretization-invariant emulators for partial-differential equations (PDEs) describing physical systems. Neural operators specifically employ deep neural networks to approximate…

We propose a method combining boundary integral equations and neural networks (BINet) to solve partial differential equations (PDEs) in both bounded and unbounded domains. Unlike existing solutions that directly operate over original PDEs,…