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Solving partial differential equations remains a central challenge in scientific machine learning. Neural operators offer a promising route by learning mappings between function spaces and enabling resolution-independent inference, yet they…

Neural Operators that directly learn mappings between function spaces, such as Deep Operator Networks (DONs) and Fourier Neural Operators (FNOs), have received considerable attention. Despite the universal approximation guarantees for DONs…

Fourier Neural Operator (FNO) is a powerful and popular operator learning method. However, FNO is mainly used in forward prediction, yet a great many applications rely on solving inverse problems. In this paper, we propose an invertible…

Solving cell problems in homogenization is hard, and available deep-learning frameworks fail to match the speed and generality of traditional computational frameworks. More to the point, it is generally unclear what to expect of…

The predictive accuracy of operator learning frameworks depends on the quality and quantity of available training data (input-output function pairs), often requiring substantial amounts of high-fidelity data, which can be challenging to…

Neural operators have emerged as powerful deep learning frameworks for approximating solution operators of parameterized partial differential equations (PDE). However, current methods predominantly rely on multilayer perceptrons (MLPs) for…

Memory complexity and data scarcity have so far prohibited learning solution operators of partial differential equations (PDEs) at high resolutions. We address these limitations by introducing a new data efficient and highly parallelizable…

This paper introduces a rigorous mathematical framework for neural network explainability, and more broadly for the explainability of equivariant operators called Group Equivariant Operators (GEOs) based on Group Equivariant Non-Expansive…

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the $i$-th neuron in a…

In this paper, we propose physics-informed neural operators (PINO) that combine training data and physics constraints to learn the solution operator of a given family of parametric Partial Differential Equations (PDE). PINO is the first…

Solving the wave equation is fundamental for geophysical applications. However, numerical solutions of the Helmholtz equation face significant computational and memory challenges. Therefore, we introduce a physics-informed convolutional…

Elliptic partial differential equations (PDEs) are a major class of time-independent PDEs that play a key role in many scientific and engineering domains such as fluid dynamics, plasma physics, and solid mechanics. Recently, neural…

In computational physics, a longstanding challenge lies in finding numerical solutions to partial differential equations (PDEs). Recently, research attention has increasingly focused on Neural Operator methods, which are notable for their…

Neural Operators (NOs) are a leading method for surrogate modeling of partial differential equations. Unlike traditional neural networks, which approximate individual functions, NOs learn the mappings between function spaces. While NOs have…

Numerically solving partial differential equations typically requires fine discretization to resolve necessary spatiotemporal scales, which can be computationally expensive. Recent advances in deep learning have provided a new approach to…

Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training…

Neural operators serve as fast, data-driven surrogates for scientific modeling but typically rely on a monolithic, single-pass inference procedure that struggles to resolve high-frequency details, a limitation known as spectral bias. We…

We have studied possible applications of a particular pseudo-differential algebra in singular analysis for the construction of fundamental solutions and Green's functions of a certain class of elliptic partial differential operators. The…

The Monte Carlo-type Neural Operator (MCNO) introduces a framework for learning solution operators of one-dimensional partial differential equations (PDEs) by directly learning the kernel function and approximating the associated integral…

Local neural operator (LNO) conception has provided a feasible way for scientific computations. The LNO learns transient partial differential equations from random field samples, and then the pre-trained LNO solves practical problems on…