相关论文: Tailored Finite Point Operator Networks for Interf…

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…

We describe a new form of diagonalization for linear two point constant coefficient differential operators with arbitrary linear boundary conditions. Although the diagonalization is in a weaker sense than that usually employed to solve…

Neural operators have achieved strong performance in learning solution operators of partial differential equations (PDEs), but their inherently continuous representations struggle to capture discontinuities and sharp transitions. Existing…

We show that the physics-informed neural networks (PINNs), in combination with some recently developed discontinuity capturing neural networks, can be applied to solve optimal control problems subject to partial differential equations…

Deep Operator Networks (DeepONets) are among the most prominent frameworks for operator learning, grounded in the universal approximation theorem for operators. However, training DeepONets typically requires significant computational…

Deep neural networks (DNNs) have demonstrated their effectiveness in a wide range of computer vision tasks, with the state-of-the-art results obtained through complex and deep structures that require intensive computation and memory.…

This paper advances the use of physics-informed neural networks (PINNs) architectures to address moving interface problems via the level set method. Originally developed for other PDE-based problems, we particularly leverage PirateNet's…

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 machine learning algorithms dedicated to solving semi-linear PDEs are improved by using different neural network architectures and different parameterizations. These algorithms are compared to a new one that solves a fixed point…

Physics-informed neural networks (PINNs) [4, 10] are an approach for solving boundary value problems based on differential equations (PDEs). The key idea of PINNs is to use a neural network to approximate the solution to the PDE and to…

In this paper, we study arbitrary order extended finite element (XFE) methods based on two discontinuous Galerkin (DG) schemes in order to solve elliptic interface problems in two and three dimensions. Optimal error estimates in the…

Deep operator network (DeepONet) has demonstrated great success in various learning tasks, including learning solution operators of partial differential equations. In particular, it provides an efficient approach to predict the evolution…

Nonlinear physical phenomena often show complex multiscale interactions; motivated by the principles of multiscale modeling in scientific computing, we propose PAS-Net, a physics-informed Adaptive-Scale Deep Operator Network for learning…

Efficiently solving constrained optimization problems is crucial for numerous real-world applications, yet traditional solvers are often computationally prohibitive for real-time use. Machine learning-based approaches have emerged as a…

We present PFNN, a penalty-free neural network method, to efficiently solve a class of second-order boundary-value problems on complex geometries. To reduce the smoothness requirement, the original problem is reformulated to a weak form so…

We present a novel deep operator network (DeepONet) architecture for operator learning, the ensemble DeepONet, that allows for enriching the trunk network of a single DeepONet with multiple distinct trunk networks. This trunk enrichment…

The recent impressive results of deep learning-based methods on computer vision applications brought fresh air to the research and industrial community. This success is mainly due to the process that allows those methods to learn…

New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is not only to get an accurate solution but also an accurate first order derivative at the interface (from each side).…

Reconstructing high-fidelity fluid flow fields from sparse sensor measurements is vital for many science and engineering applications but remains challenging because of dimensional disparities between state and observational spaces. Due to…

Deep operator networks (DeepONets, DONs) offer a distinct advantage over traditional neural networks in their ability to be trained on multi-resolution data. This property becomes especially relevant in real-world scenarios where…