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Neural networks are discrete entities: subdivided into discrete layers and parametrized by weights which are iteratively optimized via difference equations. Recent work proposes networks with layer outputs which are no longer quantized but…

We present a variant of the solver in Zepeda-N\'u\~nez and Demanet (2014), for the 2D high-frequency Helmholtz equation in heterogeneous acoustic media. By changing the domain decomposition from a layered to a grid-like partition, this…

Interface problems pose significant challenges due to the discontinuity of their solutions, particularly when they involve singular perturbations or high-contrast coefficients, resulting in intricate singularities that complicate…

Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information of the target PDEs such as…

Partial differential equations (PDEs) play a foundational role in modeling physical phenomena. This study addresses the challenging task of determining variable coefficients within PDEs from measurement data. We introduce a novel neural…

Partial differential equations (PDEs) with Dirichlet boundary conditions defined on boundaries with simple geometry have been succesfuly treated using sigmoidal multilayer perceptrons in previous works. This article deals with the case of…

Neural operators, as an efficient surrogate model for learning the solutions of PDEs, have received extensive attention in the field of scientific machine learning. Among them, attention-based neural operators have become one of the…

Wavelet-based grid adaptation methods use multiresolution analysis for error estimation, offering a mathematically rigorous approach to adaptive grid refinement when solving Partial Differential Equations (PDEs). However, applying these…

Multiphysics problems such as multicomponent diffusion, phase transformations in multiphase systems and alloy solidification involve numerical solution of a coupled system of nonlinear partial differential equations (PDEs). Numerical…

In this work, we introduce implicit Finite Operator Learning (iFOL) for the continuous and parametric solution of partial differential equations (PDEs) on arbitrary geometries. We propose a physics-informed encoder-decoder network to…

Operator learning has emerged as a promising tool for accelerating the solution of partial differential equations (PDEs). The Deep Operator Networks (DeepONets) represent a pioneering framework in this area: the "vanilla" DeepONet is valued…

An important application of neural networks to scientific computing has been the learning of non-linear operators. In this framework, a neural network is trained to fit a non-linear map between two infinite dimensional spaces, for example,…

The Helmholtz equation is fundamental to wave modeling in acoustics, electromagnetics, and seismic imaging, yet high-frequency regimes remain challenging due to the ``pollution effect''. We propose FD-MGDL, an adaptive framework integrating…

The creation of unstable heavy particles at the Large Hadron Collider is the most direct way to address some of the deepest open questions in physics. Collisions typically produce variable-size sets of observed particles which have inherent…

The recent rise of deep learning has led to numerous applications, including solving partial differential equations using Physics-Informed Neural Networks. This approach has proven highly effective in several academic cases. However, their…

We propose a novel fine-tuning method to achieve multi-operator learning through training a distributed neural operator with diverse function data and then zero-shot fine-tuning the neural network using physics-informed losses for…

We present three multi-scale similarity learning architectures, or DeepSim networks. These models learn pixel-level matching with a contrastive loss and are agnostic to the geometry of the considered scene. We establish a middle ground…

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…

Design and optimal control problems are among the fundamental, ubiquitous tasks we face in science and engineering. In both cases, we aim to represent and optimize an unknown (black-box) function that associates a performance/outcome to a…

We introduce Discontinuous Galerkin Finite Element Operator Network (DG--FEONet), a data-free operator learning framework that combines the strengths of the discontinuous Galerkin (DG) method with neural networks to solve parametric partial…