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Resolving Partial Differential Equations (PDEs) through numerical discretization methods like the Finite Element Method presents persistent challenges associated with computational complexity, despite achieving a satisfactory solution approximation. To surmount these computational hurdles, interpolation techniques are employed to precompute models offline, facilitating rapid online solutions within a metamodel. Probability distribution frameworks play a crucial role in data modeling across various fields such as physics, statistics, and machine learning. Optimal Transport (OT) has emerged as a robust approach for probability distribution interpolation due to its ability to account for spatial dependencies and continuity. However, interpolating in high-dimensional spaces encounters challenges stemming from the curse of dimensionality. The article offers insights into the application of OT, addressing associated challenges and proposing a novel methodology. This approach utilizes the distinctive arrangement of an ANOVA-based sampling to interpolate between more than two distributions using a step-by-step matching algorithm. Subsequently, the ANOVA-PGD method is employed to construct the metamodel, providing a comprehensive solution to address the complexities inherent in distribution interpolation.

Trajectory optimization is a complex process that includes an infinite number of possibilities and combinations. This work focuses on a particular aspect of the trajectory optimization, related to the optimization of a velocity along a predefined path, with the aim of minimizing power consumption. To tackle the problem, a functional formulation and minimization strategy is developed, by means of the Euler-Lagrange equation. The minimization is later performed using a neural network approach. The strategy is deemed Lagrange-Net, as it is based on the minimization of the energy functional, by the means of Lagrange's equation and neural network approximations. [ABSTRACT FROM AUTHOR]

This document is a correction notice for an article titled "A Regularized Real-Time Integrator for Data-Driven Control of Heating Channels" published in the journal Computation. The correction addresses two errors in the original publication. Firstly, the section on data availability has been updated to clarify that the data is the property of the project's consortium and cannot be made available. Secondly, there were confidential data on the y-axis of several figures, which have been replaced by a scale in the corrected figures. The authors state that these corrections do not affect the scientific conclusions of the article. [Extracted from the article]

Solving mechanical problems in large structures with rich localized behaviors remains a challenging issue despite the enormous advances in numerical procedures and computational performance. In particular, these localized behaviors need for extremely fine descriptions, and this has an associated impact in the number of degrees of freedom from one side, and the decrease of the time step employed in usual explicit time integrations, whose stability scales with the size of the smallest element involved in the mesh. In the present work we propose a data-driven technique for learning the rich behavior of a local patch and integrate it into a standard coarser description at the structure level. Thus, localized behaviors impact the global structural response without needing an explicit description of that fine scale behaviors.