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This focused review explores a range of neural operator architectures for approximating solutions to parametric partial differential equations (PDEs), emphasizing high-level concepts and practical implementation strategies. The study covers…
Learning solution operators for partial differential equations (PDEs) has become a foundational task in scientific machine learning. However, existing neural operator methods require abundant training data for each specific PDE and lack the…
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…
The scientific computation methods development in conjunction with artificial intelligence technologies remains a hot research topic. Finding a balance between lightweight and accurate computations is a solid foundation for this direction.…
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…
The classical development of neural networks has primarily focused on learning mappings between finite dimensional Euclidean spaces or finite sets. We propose a generalization of neural networks to learn operators, termed neural operators,…
Recently, machine learning methods have gained significant traction in scientific computing, particularly for solving Partial Differential Equations (PDEs). However, methods based on deep neural networks (DNNs) often lack convergence…
Neural operators learn to map initial conditions to the terminal solution of partial differential equations (PDEs), providing a surrogate for the full operator mapping. This enables rapid prediction across different input configurations.…
We present a numerical framework for deep neural network (DNN) modeling of unknown time-dependent partial differential equations (PDE) using their trajectory data. Unlike the recent work of [Wu and Xiu, J. Comput. Phys. 2020], where the…
Discovering hidden physical laws and identifying governing system parameters from sparse observations are central challenges in computational science and engineering. Existing data-driven methods, such as physics-informed neural networks…
The order/dimension of models derived on the basis of data is commonly restricted by the number of observations, or in the context of monitored systems, sensing nodes. This is particularly true for structural systems (e.g., civil or…
Neural ordinary differential equations (NODEs), one of the most influential works of the differential equation-based deep learning, are to continuously generalize residual networks and opened a new field. They are currently utilized for…
Dynamical system models such as Recurrent Neural Networks (RNNs) have become increasingly popular as hypothesis-generating tools in scientific research. Evaluating the dynamics in such networks is key to understanding their learned…
Neural differential equation models have garnered significant attention in recent years for their effectiveness in machine learning applications.Among these, fractional differential equations (FDEs) have emerged as a promising tool due to…
Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…
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…
Recent work in deep learning focuses on solving physical systems in the Ordinary Differential Equation or Partial Differential Equation. This current work proposed a variant of Convolutional Neural Networks (CNNs) that can learn the hidden…
This paper introduces a data-driven operator learning method for multiscale partial differential equations, with a particular emphasis on preserving high-frequency information. Drawing inspiration from the representation of multiscale…
Neural operators aim to approximate the solution operator of a system of differential equations purely from data. They have shown immense success in modeling complex dynamical systems across various domains. However, the occurrence of…
Coupled partial differential equations (PDEs) are key tasks in modeling the complex dynamics of many physical processes. Recently, neural operators have shown the ability to solve PDEs by learning the integral kernel directly in…