Related papers: Manifold learning techniques and model reduction a…
Supervised manifold learning methods learn data representations by preserving the geometric structure of data while enhancing the separation between data samples from different classes. In this work, we propose a theoretical study of…
In this paper, we study deep neural networks (DNNs) for solving high-dimensional evolution equations with oscillatory solutions. Different from deep least-squares methods that deal with time and space variables simultaneously, we propose a…
Probabilistic Manifold Decomposition (PMD)\cite{doi:10.1137/25M1738863}, developed in our earlier work, provides a nonlinear model reduction by embedding high-dimensional dynamics onto low-dimensional probabilistic manifolds. The PMD has…
We develop new adaptive algorithms for temporal integration of nonlinear evolution equations on tensor manifolds. These algorithms, which we call step-truncation methods, are based on performing one time step with a conventional…
Several nonlinear model reduction techniques are compared for the three cases of the non-parallel version of the Kuramoto-Sivashinsky equation, the transient regime of flow past a cylinder at $Re=100$ and fully developed flow past a…
Leveraging nonlinear parametrizations for model reduction can overcome the Kolmogorov barrier that affects transport-dominated problems. In this work, we build on the reduced dynamics given by Neural Galerkin schemes and propose to…
We present a model reduction approach for the real-time solution of time-dependent nonlinear partial differential equations (PDEs) with parametric dependencies. The approach integrates several ingredients to develop efficient and accurate…
Even for the gradient descent (GD) method applied to neural network training, understanding its optimization dynamics, including convergence rate, iterate trajectories, function value oscillations, and especially its implicit acceleration,…
The purpose of this research work is to employ the Optimal Auxiliary Function Method (OAFM) for obtaining numerical approximations of time-dependent nonlinear partial differential equations (PDEs) that arise in many disciplines of science…
The nonlinear coupled reaction-diffusion (NCRD) systems are important in the formation of spatiotemporal patterns in many scientific and engineering fields, including physical and chemical processes, biology, electrochemical processes,…
Neural networks are increasingly recognized as a powerful numerical solution technique for partial differential equations (PDEs) arising in diverse scientific computing domains, including quantum many-body physics. In the context of…
Over the past few years, neural network methods have evolved in various directions for approximating partial differential equations (PDEs). A promising new development is the integration of neural networks with classical numerical…
Simulating the time evolution of Partial Differential Equations (PDEs) of large-scale systems is crucial in many scientific and engineering domains such as fluid dynamics, weather forecasting and their inverse optimization problems.…
In this paper, we put forth an evolve-then-correct reduced order modeling approach that combines intrusive and nonintrusive models to take hidden physical processes into account. Specifically, we split the underlying dynamics into known and…
Partial differential equations (PDEs) form a central component of scientific computing. Among recent advances in deep learning, evolutionary neural networks have been developed to successively capture the temporal dynamics of time-dependent…
We propose a novel type of nonlinear solver acceleration for systems of nonlinear partial differential equations (PDEs) that is based on online/adaptive learning. It is applied in the context of multiphase flow in porous media. The proposed…
A method for model reduction in nonlinear ODE systems is demonstrated through computational examples. The method does not require an implicit separation of time-scales in the fine dynamics to be effective. From the computational standpoint,…
We consider a fast-slow partial differential equation (PDE) with reaction-diffusion dynamics in the fast variable and the slow variable driven by a differential operator on a bounded domain. Assuming a transcritical normal form for the…
We introduce generative models for accelerating simulations of complex systems through learning and evolving their effective dynamics. In the proposed Generative Learning of Effective Dynamics (G-LED), instances of high dimensional data are…
Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations (PDEs). Naturally, reduced-order modeling techniques come at the price of computational accuracy for a decrease in computation…