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Neural operators are becoming the default tools to learn solutions to governing partial differential equations (PDEs) in weather and ocean forecasting applications. Despite early promising achievements, significant challenges remain,…

Partial differential equations (PDEs) govern a wide variety of dynamical processes in science and engineering, yet obtaining their numerical solutions often requires high-resolution discretizations and repeated evaluations of complex…

Neural PDE solvers are increasingly used as learned surrogates for families of partial differential equations, where the key machine learning challenge is not only interpolation on a fixed benchmark distribution but generalization under…

Modeling high-frequency information is a critical challenge in scientific machine learning. For instance, fully turbulent flow simulations of the Navier-Stokes equations at Reynolds numbers 3500 and above can generate high-frequency signals…

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…

Neural operators have emerged as a powerful, data-driven paradigm for learning solution operators of partial differential equations (PDEs). State-of-the-art architectures, such as the Fourier Neural Operator (FNO), have achieved remarkable…

Operator learning has emerged as a promising paradigm for developing efficient surrogate models to solve partial differential equations (PDEs). However, existing approaches often overlook the domain knowledge inherent in the underlying PDEs…

The use of neural operators in a digital twin model of an offshore floating structure can provide a paradigm shift in structural response prediction and health monitoring, providing valuable information for real-time control. In this work,…

Fourier Neural Operators (FNO) have emerged as promising solutions for efficiently solving partial differential equations (PDEs) by learning infinite-dimensional function mappings through frequency domain transformations. However, the…

Data-driven machine learning approaches are being increasingly used to solve partial differential equations (PDEs). They have shown particularly striking successes when training an operator, which takes as input a PDE in some family, and…

We consider solving partial differential equations (PDEs) with Fourier neural operators (FNOs), which operate in the frequency domain. Since the laws of physics do not depend on the coordinate system used to describe them, it is desirable…

Fourier Neural Operators (FNO) offer a principled approach to solving challenging partial differential equations (PDE) such as turbulent flows. At the core of FNO is a spectral layer that leverages a discretization-convergent representation…

Recent works have shown that traditional Neural Network (NN) architectures display a marked frequency bias in the learning process. Namely, the NN first learns the low-frequency features before learning the high-frequency ones. In this…

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…

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…

Neural Operators (NOs) are machine learning models designed to solve partial differential equations (PDEs) by learning to map between function spaces. Neural Operators such as the Deep Operator Network (DeepONet) and the Fourier Neural…

Traditionally, neural networks have been employed to learn the mapping between finite-dimensional Euclidean spaces. However, recent research has opened up new horizons, focusing on the utilization of deep neural networks to learn operators…

Neural operators can learn nonlinear mappings between function spaces and offer a new simulation paradigm for real-time prediction of complex dynamics for realistic diverse applications as well as for system identification in science and…

Learning the mapping between two function spaces has garnered considerable research attention. However, learning the solution operator of partial differential equations (PDEs) remains a challenge in scientific computing. Fourier neural…

Solving partial differential equations (PDEs) by neural networks as well as Kolmogorov-Arnold Networks (KANs), including physics-informed neural networks (PINNs), physics-informed KANs (PIKANs), and neural operators, are known to exhibit…