Related papers: Deep Convolutional Ritz Method: Parametric PDE sur…

Recently deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited to bounded domains. In this work, we…

Polynomial chaos expansion (PCE) is a powerful surrogate model-based reliability analysis method. Generally, a PCE model with a higher expansion order is usually required to obtain an accurate surrogate model for some complex non-linear…

In recent years, increasingly complex computational models are being built to describe physical systems which has led to increased use of surrogate models to reduce computational cost. In problems related to Structural Health Monitoring…

Neural surrogates for partial differential equations (PDEs) have become popular due to their potential to quickly simulate physics. With a few exceptions, neural surrogates generally treat the forward evolution of time-dependent PDEs as a…

The study of parametric differential equations plays a crucial role in weather forecasting and epidemiological modeling. These phenomena are better represented using fractional derivatives due to their inherent memory or hereditary effects.…

Derivation of the probability density evolution provides invaluable insight into the behavior of many stochastic systems and their performance. However, for most real-time applica-tions, numerical determination of the probability density…

Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…

Physics informed neural networks (PINNs) have emerged as a powerful tool to provide robust and accurate approximations of solutions to partial differential equations (PDEs). However, PINNs face serious difficulties and challenges when…

Neural networks (NNs) have gained significant attention across various engineering disciplines, particularly in design optimization, where they are used to build surrogate models for high-dimensional regression problems. Despite their power…

Physics-informed neural networks (PINNs) have successfully addressed various computational physics problems based on partial differential equations (PDEs). However, while tackling issues related to irregularities like singularities and…

Identifying parameters in partial differential equations (PDEs) represents a very broad class of applied inverse problems. In recent years, several unsupervised learning approaches using (deep) neural networks have been developed to solve…

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.…

While recent advances in deep learning have shown promising efficiency gains in solving time-dependent partial differential equations (PDEs), matching the accuracy of conventional numerical solvers still remains a challenge. One strategy to…

Solving partial differential equations (PDEs) is an important research means in the fields of physics, biology, and chemistry. As an approximate alternative to numerical methods, PINN has received extensive attention and played an important…

Transformer models are increasingly used for solving Partial Differential Equations (PDEs). Several adaptations have been proposed, all of which suffer from the typical problems of Transformers, such as quadratic memory and time complexity.…

In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating PDEs with two characteristic scales. From a continuous perspective, our formulation…

This article presents a reduced-order modeling methodology via deep convolutional neural networks (CNNs) for shape optimization applications. The CNN provides a nonlinear mapping between the shapes and their associated attributes while…

Implicit Neural Representations (INRs) have emerged as promising surrogates for large 3D scientific simulations due to their ability to continuously model spatial and conditional fields, yet they face a critical fidelity-speed dilemma: deep…

We examine the challenges associated with numerical integration when applying Neural Networks to solve Partial Differential Equations (PDEs). We specifically investigate the Deep Ritz Method (DRM), chosen for its practical applicability and…

In various engineering and applied science applications, repetitive numerical simulations of partial differential equations (PDEs) for varying input parameters are often required (e.g., aircraft shape optimization over many design…