相关论文: Neural Operators Meet Energy-based Theory: Operato…

We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs), with a high degree of accuracy and up to a desired tolerance. We develop a differentiable…

Neural Ordinary Differential Equations (NODEs) have proven to be a powerful modeling tool for approximating (interpolation) and forecasting (extrapolation) irregularly sampled time series data. However, their performance degrades…

Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs). However, purely data-driven models often require extensive training data and can generalize poorly, especially in small-data…

Physics-informed neural networks (PINNs) have been proven as a promising way for solving various partial differential equations, especially high-dimensional ones and those with irregular boundaries. However, their capabilities in real…

This study presents a novel unsupervised convolutional Neural Network (NN) architecture with nonlocal interactions for solving Partial Differential Equations (PDEs). The nonlocal Peridynamic Differential Operator (PDDO) is employed as a…

DeepONet has recently been proposed as a representative framework for learning nonlinear mappings between function spaces. However, when it comes to approximating solution operators of partial differential equations (PDEs) with…

Conservation of energy is at the core of many physical phenomena and dynamical systems. There have been a significant number of works in the past few years aimed at predicting the trajectory of motion of dynamical systems using neural…

Attention mechanisms have emerged as transformative tools in core AI domains such as natural language processing and computer vision. Yet, their largely untapped potential for modeling intricate physical systems presents a compelling…

Discovering hidden partial differential equations (PDEs) and operators from data is an important topic at the frontier between machine learning and numerical analysis. This doctoral thesis introduces theoretical results and deep learning…

Neural operators (NOs) have emerged as effective tools for modeling complex physical systems in scientific machine learning. In NOs, a central characteristic is to learn the governing physical laws directly from data. In contrast to other…

Operator learning has emerged as a powerful tool in scientific computing for approximating mappings between infinite-dimensional function spaces. A primary application of operator learning is the development of surrogate models for the…

We introduce a method that combines neural operators, physics-informed machine learning, and standard numerical methods for solving PDEs. The proposed approach extends each of the aforementioned methods and unifies them within a single…

Although there is a substantial body of literature on control and optimization problems for parabolic and hyperbolic systems, the specific problem of controlling and optimizing the coefficients of the associated operators within such…

Neural operators have emerged as powerful tools for learning solution operators of partial differential equations. However, in time-dependent problems, standard training strategies such as teacher forcing introduce a mismatch between…

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,…

Physics-informed neural networks (PINNs) are a newly emerging research frontier in machine learning, which incorporate certain physical laws that govern a given data set, e.g., those described by partial differential equations (PDEs), into…

This work formulates a new approach to reduced modeling of parameterized, time-dependent partial differential equations (PDEs). The method employs Operator Inference, a scientific machine learning framework combining data-driven learning…

The precise simulation of turbulent flows is of immense importance in a variety of scientific and engineering fields, including climate science, freshwater science, and the development of energy-efficient manufacturing processes. Within the…

Modelling complex multiphysics systems governed by nonlinear and strongly coupled partial differential equations (PDEs) is a cornerstone in computational science and engineering. However, it remains a formidable challenge for traditional…

Physics-informed neural networks (PINNs) have emerged as a promising approach to solving partial differential equations (PDEs) using neural networks, particularly in data-scarce scenarios, due to their unsupervised training capability.…