相关论文: Render unto Numerics: Orthogonal Polynomial Neural…
Neural operators have been validated as promising deep surrogate models for solving partial differential equations (PDEs). Despite the critical role of boundary conditions in PDEs, however, only a limited number of neural operators robustly…
Designing universal artificial intelligence (AI) solver for partial differential equations (PDEs) is an open-ended problem and a significant challenge in science and engineering. Currently, data-driven solvers have achieved great success,…
In this paper, we introduce Proper Orthogonal Decomposition Neural Operators (PODNO) for solving partial differential equations (PDEs) dominated by high-frequency components. Building on the structure of Fourier Neural Operators (FNO),…
Neural operators have emerged as powerful surrogates for the solution of partial differential equations (PDEs), yet their ability to handle general, highly variable boundary conditions (BCs) remains limited. Existing approaches often fail…
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…
Boundary conditions (BCs) are important groups of physics-enforced constraints that are necessary for solutions of Partial Differential Equations (PDEs) to satisfy at specific spatial locations. These constraints carry important physical…
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…
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…
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…
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 (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…
Neural operators (NO) are discretization invariant deep learning methods with functional output and can approximate any continuous operator. NO have demonstrated the superiority of solving partial differential equations (PDEs) over other…
Partial differential equations (PDEs) govern a wide range of physical phenomena, but their numerical solution remains computationally demanding, especially when repeated simulations are required across many parameter settings. Recent…
For linear partial differential equations with known fundamental solutions, this work introduces a novel operator learning framework that relies exclusively on domain boundary data, including solution values and normal derivatives, rather…
Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the…
Neural operator (NO) architectures learn nonlinear maps between infinite-dimensional function spaces and are widely used to accelerate simulation and enable data-driven model discovery. While universality results ensure expressivity, they…
In this paper, we propose Neumann Series Neural Operator (NSNO) to learn the solution operator of Helmholtz equation from inhomogeneity coefficients and source terms to solutions. Helmholtz equation is a crucial partial differential…
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…
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 Monte Carlo-type Neural Operator (MCNO) introduces a lightweight architecture for learning solution operators for parametric PDEs by directly approximating the kernel integral using a Monte Carlo approach. Unlike Fourier Neural…