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Numerical solutions of partial differential equations (PDEs) require expensive simulations, limiting their application in design optimization, model-based control, and large-scale inverse problems. Surrogate modeling techniques seek to…
In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based…
Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost function of a Neural Network. Most…
We enhance machine learning algorithms for learning model parameters in complex systems represented by ordinary differential equations (ODEs) with domain decomposition methods. The study evaluates the performance of two approaches, namely…
Addressing high-dimensional partial differential equations to derive effective actions within the functional renormalization group is formidable, especially when considering various field configurations, including inhomogeneous states, even…
The geometry of generative models serves as the basis for interpolation, model inspection, and more. Unfortunately, most generative models lack a principal notion of geometry without restrictive assumptions on either the model or the data…
Traditional Monte Carlo integration using uniform random sampling exhibits degraded efficiency in low-regularity or high-dimensional problems. We propose a novel deep learning framework based on deterministic number-theoretic sampling…
Content and image generation consist in creating or generating data from noisy information by extracting specific features such as texture, edges, and other thin image structures. We are interested here in generative models, and two main…
In recent years, neural networks have achieved remarkable progress in various fields and have also drawn much attention in applying them on scientific problems. A line of methods involving neural networks for solving partial differential…
Traditional projection-based reduced-order modeling approximates the full-order model by projecting it onto a linear subspace. With a fast-decaying Kolmogorov $n$-width of the solution manifold, the resulting reduced-order model (ROM) can…
In this study, we present and validate the predictive capability of the Physics-Informed Neural Networks (PINNs) methodology for solving a variety of engineering and biological dynamical systems governed by ordinary differential equations…
Complex physical systems are often described by partial differential equations (PDEs) that depend on parameters such as the Reynolds number in fluid mechanics. In applications such as design optimization or uncertainty quantification,…
This research presents the development of an innovative algorithm tailored for the adaptive sampling of residual points within the framework of Physics-Informed Neural Networks (PINNs). By addressing the limitations inherent in existing…
We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) solved by the method of lines and their representation in chaotic ordinary differential equations, based on neural ordinary…
Deep Learning-based Reduced Order Models (DL-ROMs) provide nowadays a well-established class of accurate surrogate models for complex physical systems described by parametrized PDEs, by nonlinearly compressing the solution manifold into a…
This work investigates a two-stage method for constructing projection-based reduced-order models (ROMs) of parameterized partial differential equations (PDEs). Based on established tensorial ROM methodology, the proposed approach reduces…
In this paper, we consider approximating the parameter-to-solution maps of parametric partial differential equations (PPDEs) using deep neural networks (DNNs). We propose an efficient approach combining reduced collocation methods (RCMs)…
Numerically solving high-dimensional partial differential equations (PDEs) is a major challenge. Conventional methods, such as finite difference methods, are unable to solve high-dimensional PDEs due to the curse-of-dimensionality. A…
Quantum physics-informed neural networks (QPINNs) have recently emerged as a promising framework for the solution of partial differential equations (PDEs), with several studies reporting improved convergence and accuracy relative to…
Physics-informed neural networks (PINNs) are a versatile tool in the burgeoning field of scientific machine learning for solving partial differential equations (PDEs). However, determining suitable training strategies for them is not…