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A general, variational approach to derive low-order reduced systems is presented. The approach is based on the concept of optimal parameterizing manifold (OPM) that substitutes the more classical notions of invariant or slow manifold when…

A promising approach to investigating high-dimensional problems is to identify their intrinsically low-dimensional features, which can be achieved through recently developed techniques for effective low-dimensional representation of…

Systems may depend on parameters which one may control, or which serve to optimise the system, or are imposed externally, or they could be uncertain. This last case is taken as the ``Leitmotiv'' for the following. A reduced order model is…

In recent years, methods of approximate parameter estimation have attracted considerable interest in complex problems where exact likelihoods are hard to obtain. In their most basic form, Bayesian methods such as Approximate Bayesian…

Latent variable models represent a useful tool for the analysis of complex data when the constructs of interest are not observable. A problem related to these models is that the integrals involved in the likelihood function cannot be solved…

This article deals with the adaptive and approximative computation of the Lam\'e equations. The equations of linear elasticity are considered as boundary integral equations and solved in the setting of the boundary element method (BEM).…

Running agent-based models (ABMs) is a burdensome computational task, specially so when considering the flexibility ABMs intrinsically provide. This paper uses a bundle of model configuration parameters along with obtained results from a…

Scientific machine learning increasingly uses spectral methods to understand physical systems. Current spectral learning approaches provide only point estimates without uncertainty quantification, limiting their use in safety-critical…

We consider how mathematical models enable predictions for conditions that are qualitatively different from the training data. We propose techniques based on information topology to find models that can apply their learning in regimes for…

Understanding the macroscopic characteristics of biological complexes demands precision and specificity in statistical ensemble modeling. One of the primary challenges in this domain lies in sampling from particular subsets of the…

Manifold alignment is a type of data fusion technique that creates a shared low-dimensional representation of data collected from multiple domains, enabling cross-domain learning and improved performance in downstream tasks. This paper…

Discrimination mitigation within machine learning (ML) models could be complicated because multiple factors may be interwoven hierarchically and historically. Yet few existing fairness measures can capture the discrimination level within ML…

A methodology for using random sketching in the context of model order reduction for high-dimensional parameter-dependent systems of equations was introduced in [Balabanov and Nouy 2019, Part I]. Following this framework, we here construct…

We present a new approach for nonlinear dimensionality reduction, specifically designed for computationally expensive mathematical models. We leverage autoencoders to discover a one-dimensional neural active manifold (NeurAM) capturing the…

When gradient-based methods are impractical, black-box optimization (BBO) provides a valuable alternative. However, BBO often struggles with high-dimensional problems and limited trial budgets. In this work, we propose a novel approach…

Model reduction of large nonlinear systems often involves the projection of the governing equations onto linear subspaces spanned by carefully-selected modes. The criteria to select the modes relevant for reduction are usually…

Parametrizations of data manifolds in shape spaces can be computed using the rich toolbox of Riemannian geometry. This, however, often comes with high computational costs, which raises the question if one can learn an efficient neural…

In this paper, we consider the problem of manifold approximation with affine subspaces. Our objective is to discover a set of low dimensional affine subspaces that represents manifold data accurately while preserving the manifold's…

Chemical kinetic models in terms of ordinary differential equations correspond to finite dimensional dissipative dynamical systems involving a multiple time scale structure. Most dimension reduction approaches aimed at a slow…

Recently, diffusion models have been used to solve various inverse problems in an unsupervised manner with appropriate modifications to the sampling process. However, the current solvers, which recursively apply a reverse diffusion step…