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Reduced Order Models (ROMs) form essential tools across engineering domains by virtue of their function as surrogates for computationally intensive digital twinning simulators. Although purely data-driven methods are available for ROM…

Partial differential equations (PDEs) play a central role in describing many physical phenomena. Various scientific and engineering applications demand a versatile and differentiable PDE solver that can quickly generate solutions with…

The Kolmogorov $n$-width of the solution manifolds of transport-dominated problems can decay slowly. As a result, it can be challenging to design efficient and accurate reduced order models (ROMs) for such problems. To address this issue,…

Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be…

We introduce a method for the fast numerical approximation of linear, second-order parabolic partial differential equations (PDEs for short) with time-independent coefficients based on model order reduction techniques and the Laplace…

Nearly all model-reduction techniques project the governing equations onto a linear subspace of the original state space. Such subspaces are typically computed using methods such as balanced truncation, rational interpolation, the…

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…

Solving partial differential equations (PDEs) is a required step in the simulation of natural and engineering systems. The associated computational costs significantly increase when exploring various scenarios, such as changes in initial or…

Linear reduced-order modeling (ROM) simplifies complex simulations by approximating the behavior of a system using a simplified kinematic representation. Typically, ROM is trained on input simulations created with a specific spatial…

In order to investigate correspondences between 3D shapes, many methods rely on a feature descriptor which is invariant under almost isometric transformations. An interesting class of models for such descriptors relies on partial…

Linear reduced-order modeling (ROM) is widely used for efficient simulation of deformation dynamics, but its accuracy is often limited by the fixed linearization of the reduced mapping. We propose a new adaptive strategy for linear ROM that…

In this paper, we propose an equation-based parametric Reduced Order Model (ROM), whose accuracy is improved with data-driven terms added into the reduced equations. These additions have the aim of reintroducing contributions that in…

This work proposes an autoencoder neural network as a non-linear generalization of projection-based methods for solving Partial Differential Equations (PDEs). The proposed deep learning architecture presented is capable of generating the…

Proper orthogonal decomposition (POD) allows reduced-order modeling of complex dynamical systems at a substantial level, while maintaining a high degree of accuracy in modeling the underlying dynamical systems. Advances in machine learning…

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…

Trajectory-wise data-driven reduced order models (ROMs) tend to be sensitive to training data, and thus lack robustness. We propose to construct a robust stochastic ROM closure (S-ROM) from data consisting of multiple trajectories from…

In this paper, we consider the problem of learning prediction models for spatiotemporal physical processes driven by unknown partial differential equations (PDEs). We propose a deep learning framework that learns the underlying dynamics and…

We present a deep learning emulator for stochastic and chaotic spatio-temporal systems, explicitly conditioned on the parameter values of the underlying partial differential equations (PDEs). Our approach involves pre-training the model on…

The basis generation in reduced order modeling usually requires multiple high-fidelity large-scale simulations that could take a huge computational cost. In order to accelerate these numerical simulations, we introduce a FOM/ROM hybrid…

A common strategy for the dimensionality reduction of nonlinear partial differential equations relies on the use of the proper orthogonal decomposition (POD) to identify a reduced subspace and the Galerkin projection for evolving dynamics…