相关论文: Wiener Chaos Expansion based Neural Operator for S…

Stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) are fundamental for modeling stochastic dynamics across the natural sciences and modern machine learning. Learning their solution operators with…

This study addresses the inverse problem of parameter estimation for Stochastic Differential Equations (SDEs) by minimizing a regularized discrepancy functional via Stochastic Gradient Descent (SGD). To achieve computational efficiency, we…

In this paper, we solve stochastic partial differential equations (SPDEs) numerically by using (possibly random) neural networks in the truncated Wiener chaos expansion of their corresponding solution. Moreover, we provide some…

Building surrogate models with uncertainty quantification capabilities is essential for many engineering applications where randomness, such as variability in material properties, is unavoidable. Polynomial Chaos Expansion (PCE) is widely…

A new method is described for constructing a generalized solution for stochastic differential equations. The method is based on the Cameron-Martin version of the Wiener Chaos expansion and provides a unified framework for the study of…

We study the problem of learning the law of linear stochastic partial differential equations (SPDEs) with additive Gaussian forcing from spatiotemporal observations. Most existing deep learning approaches either assume access to the driving…

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…

Neural Operators (NOs) provide a powerful framework for computations involving physical laws that can be modelled by (integro-) partial differential equations (PDEs), directly learning maps between infinite-dimensional function spaces that…

Neural operators (NOs) provide a new paradigm for efficiently solving partial differential equations (PDEs), but their training depends on costly high-fidelity data from numerical solvers, limiting applications in complex systems. We…

Numerical computations based on the Wiener Chaos Expansion (WCE) are carried out to approximate the solutions of the stochastic generalized Kuramoto--Sivashinsky (SgKS) equation driven by Brownian motion forcing. In the assessment of the…

We propose a Weak-form Physics-Informed Neural Operator (WINO), a data-free framework that combines the efficiency of neural operators with the geometric flexibility of the $\varphi$-finite element method ($\varphi$-FEM). $\varphi$-FEM is…

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…

A novel approach to approximate solutions of Stochastic Differential Equations (SDEs) by Deep Neural Networks is derived and analysed. The architecture is inspired by the notion of Deep Operator Networks (DeepONets), which is based on…

Deep neural operators are recognized as an effective tool for learning solution operators of complex partial differential equations (PDEs). As compared to laborious analytical and computational tools, a single neural operator can predict…

A novel and efficient algorithm based on the Wiener chaos expansion is proposed for the stochastic Maxwell equations driven by Wiener process. The proposed algorithm can reduce the original stochastic system to the deterministic case and…

Operator learning (OL) has emerged as a powerful tool in scientific machine learning (SciML) for approximating mappings between infinite-dimensional functional spaces. One of its main applications is learning the solution operator of…

In this paper, we propose a way to solve partial differential equations (PDEs) by combining machine learning techniques and the finite element method called Phi-FEM. For that, we use the Fourier Neural Operator (FNO), a learning mapping…

Solving partial differential equations (PDEs) is a required step in the simulation of natural and engineering systems. The associated computational costs significantly increase when exploring various scenarios, such as changes in initial or…

We propose a new class of physics-informed neural networks, called physics-informed Variational Autoencoder (PI-VAE), to solve stochastic differential equations (SDEs) or inverse problems involving SDEs. In these problems the governing…

The finite element method (FEM) is a well-established numerical method for solving partial differential equations (PDEs). However, its mesh-based nature gives rise to substantial computational costs, especially for complex multiscale…