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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…

We introduce the Laplace neural operator (LNO), which leverages the Laplace transform to decompose the input space. Unlike the Fourier Neural Operator (FNO), LNO can handle non-periodic signals, account for transient responses, and exhibit…

High-fidelity simulations of laser welding capture complex thermo-fluid phenomena, including phase change, free-surface deformation, and keyhole dynamics, however their computational cost limits large-scale process exploration and real-time…

Weather and climate data are often available at limited temporal resolution, either due to storage limitations, or in the case of weather forecast models based on deep learning, their inherently long time steps. The coarse temporal…

Neural operators learn mappings from function-dependent inputs to solutions, providing an effective framework for solving partial differential equations (PDEs). For time-dependent PDEs, existing methods typically perform long-horizon…

Numerical solvers for PDEs often struggle to balance computational cost with accuracy, especially in multiscale and time-dependent systems. Neural operators offer a promising way to accelerate simulations, but their practical deployment is…

Accurate and efficient solutions of spatiotemporal partial differential equations (PDEs), such as phase-field models, are fundamental for understanding interfacial dynamics and microstructural evolution in materials science and fluid…

The UNet-enhanced Fourier Neural Operator (UFNO) extends the Fourier Neural Operator (FNO) by incorporating a parallel UNet pathway, enabling the retention of both high- and low-frequency components. While UFNO improves predictive accuracy…

Fast and accurate predictions of turbulent flows are of great importance in the science and engineering field. In this paper, we investigate the implicit U-Net enhanced Fourier neural operator (IUFNO) in the stable prediction of long-time…

Multiscale problems are ubiquitous in physics. Numerical simulations of such problems by solving partial differential equations (PDEs) at high resolution are computationally too expensive for many-query scenarios, such as uncertainty…

High-frequency features are critical in multiscale phenomena such as turbulent flows and phase transitions, since they encode essential physical information. The recently proposed Wavelet Neural Operator (WNO) utilizes wavelets'…

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 solution of partial differential equations (PDEs) plays a central role in numerous applications in science and engineering, particularly those involving multiphase flow in porous media. Complex, nonlinear systems govern these problems…

Soil and groundwater contamination is a pervasive problem at thousands of locations across the world. Contaminated sites often require decades to remediate or to monitor natural attenuation. Climate change exacerbates the long-term site…

Laplace Neural Operators (LNOs) have recently emerged as a promising approach in scientific machine learning due to the ability to learn nonlinear maps between functional spaces. However, this framework often requires substantial amounts of…

Neural operators, which aim to approximate mappings between infinite-dimensional function spaces, have been widely applied in the simulation and prediction of physical systems. However, the limited representational capacity of network…

Exploring the outer atmosphere of the sun has remained a significant bottleneck in astrophysics, given the intricate magnetic formations that significantly influence diverse solar events. Magnetohydrodynamics (MHD) simulations allow us to…

Constitutive modeling based on continuum mechanics theory has been a classical approach for modeling the mechanical responses of materials. However, when constitutive laws are unknown or when defects and/or high degrees of heterogeneity are…

The purpose of the current work is the development of a so-called physics-encoded Fourier neural operator (PeFNO) for surrogate modeling of the quasi-static equilibrium stress field in solids. Rather than accounting for constraints from…

We present approximation theories and efficient training methods for derivative-informed Fourier neural operators (DIFNOs) with applications to PDE-constrained optimization. A DIFNO is an FNO trained by minimizing its prediction error…