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Finite element simulations have been used to solve various partial differential equations (PDEs) that model physical, chemical, and biological phenomena. The resulting discretized solutions to PDEs often do not satisfy requisite physical…

We propose a novel gray-box modeling algorithm for physical systems governed by stochastic differential equations (SDE). The proposed approach, referred to as the Deep Physics Corrector (DPC), blends approximate physics represented in terms…

Numerical solvers of Partial Differential Equations (PDEs) are of fundamental significance to science and engineering. To date, the historical reliance on legacy techniques has circumscribed possible integration of big data knowledge and…

Differential equations are widely used to describe complex dynamical systems with evolving parameters in nature and engineering. Effectively learning a family of maps from the parameter function to the system dynamics is of great…

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…

Forecasting physical signals in long time range is among the most challenging tasks in Partial Differential Equations (PDEs) research. To circumvent limitations of traditional solvers, many different Deep Learning methods have been…

This work presents a deep learning-based framework for the solution of partial differential equations on complex computational domains described with computer-aided design tools. To account for the underlying distribution of the training…

In this paper, we show a physics-informed neural network solver for the time-dependent surface PDEs. Unlike the traditional numerical solver, no extension of PDE and mesh on the surface is needed. We show a simplified prior estimate of the…

Partial differential equations (PDEs) are typically used as models of physical processes but are also of great interest in PDE-based image processing. However, when it comes to their use in imaging, conventional numerical methods for…

We investigate numerous structural connections between numerical algorithms for partial differential equations (PDEs) and neural architectures. Our goal is to transfer the rich set of mathematical foundations from the world of PDEs to…

The incompressible magnetohydrodynamic (MHD) equations are fundamental in many scientific and engineering applications. However, their strong nonlinearity and dual divergence-free constraints make them highly challenging for conventional…

Inverse problems involving partial differential equations (PDEs) with discontinuous coefficients are fundamental challenges in modeling complex spatiotemporal systems with heterogeneous structures and uncertain dynamics. Traditional…

Operator learning has emerged as a promising paradigm for developing efficient surrogate models to solve partial differential equations (PDEs). However, existing approaches often overlook the domain knowledge inherent in the underlying PDEs…

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.…

In many scientific fields, the generation and evolution of data are governed by partial differential equations (PDEs) which are typically informed by established physical laws at the macroscopic level to describe general and predictable…

Many physical processes such as weather phenomena or fluid mechanics are governed by partial differential equations (PDEs). Modelling such dynamical systems using Neural Networks is an active research field. However, current methods are…

Quantum algorithms for partial differential equations (PDEs) face severe practical constraints on near-term hardware: limited qubit counts restrict spatial resolution to coarse grids, while circuit depth limitations prevent accurate…

There have been growing interests in leveraging experimental measurements to discover the underlying partial differential equations (PDEs) that govern complex physical phenomena. Although past research attempts have achieved great success…

This paper introduces a new Convolutional Neural Network (ConvNet) architecture inspired by a class of partial differential equations (PDEs) called quasi-linear hyperbolic systems. With comparable performance on the image classification…

Partial differential equations (PDEs) govern diverse physical phenomena, yet high-fidelity numerical solutions are computationally expensive and Machine Learning approaches lack generalization. While Scientific Foundation Models (SFMs) aim…