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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…
Microstructural evolution, particularly grain growth, plays a critical role in shaping the physical, optical, and electronic properties of materials. Traditional phase-field modeling accurately simulates these phenomena but is…
Solving Singularly Perturbed Differential Equations (SPDEs) poses computational challenges arising from the rapid transitions in their solutions within thin regions. The effectiveness of deep learning in addressing differential equations…
We present a Fourier neural network (FNN) that can be mapped directly to the Fourier decomposition. The choice of activation and loss function yields results that replicate a Fourier series expansion closely while preserving a…
Physics-Informed Neural Operators provide efficient, high-fidelity simulations for systems governed by partial differential equations (PDEs). However, most existing studies focus only on multi-scale, multi-physics systems within a single…
This paper introduces the Kernel Neural Operator (KNO), a provably convergent operator-learning architecture that utilizes compositions of deep kernel-based integral operators for function-space approximation of operators (maps from…
In solving partial differential equations (PDEs), Fourier Neural Operators (FNOs) have exhibited notable effectiveness. However, FNO is observed to be ineffective with large Fourier kernels that parameterize more frequencies. Current…
Fourier Neural Operators (FNOs) have been promoted as fast, mesh-invariant surrogates for partial-differential equation solvers, with seismic studies reporting orders-of-magnitude speedup over classical methods. We revisit those claims by…
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…
Time-periodic quantum systems exhibit a rich variety of far-from-equilibrium phenomena and serve as ideal platforms for quantum engineering and control. However, simulating their dynamics with conventional numerical methods remains…
Neural Operators (NOs) are a powerful deep learning framework designed to learn the solution operator that arise from partial differential equations. This study investigates NOs ability to capture the stiff spatio-temporal dynamics of the…
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 maps between function spaces with a strong inductive bias is a central challenge in soft computing, especially when training data are scarce and standard deep architectures overfit. We introduce a \emph{neural integral operator}…
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…
Neural operators have emerged as powerful data-driven surrogates for learning solution operators of parametric partial differential equations (PDEs). However, widely used Fourier Neural Operators (FNOs) rely on global Fourier…
Fourier neural operators (FNOs) provide a mesh-independent way to learn solution operators for partial differential equations, yet their efficacy for magnetized turbulence is largely unexplored. Here we train an FNO surrogate for the 2-D…
We introduce DiffFNO, a novel diffusion framework for arbitrary-scale super-resolution strengthened by a Weighted Fourier Neural Operator (WFNO). Mode Rebalancing in WFNO effectively captures critical frequency components, significantly…
Neural operators approximate PDE solution maps, but they need not respect the symmetries of the governing equation. In out-of-distribution (OOD) regimes, a standard neural operator must often learn coordinate alignment and physical…
A large class of inverse problems for PDEs are only well-defined as mappings from operators to functions. Existing operator learning frameworks map functions to functions and need to be modified to learn inverse maps from data. We propose a…