@article {2022,
	title = {The Neural Network shifted-proper orthogonal decomposition: A machine learning approach for non-linear reduction of hyperbolic equations},
	journal = {Computer Methods in Applied Mechanics and Engineering},
	volume = {392},
	year = {2022},
	abstract = {<p>Models with dominant advection always posed a difficult challenge for projection-based reduced order modelling. Many methodologies that have recently been proposed are based on the pre-processing of the full-order solutions to accelerate the Kolmogorov N-width decay thereby obtaining smaller linear subspaces with improved accuracy. These methods however must rely on the knowledge of the characteristic speeds in phase space of the solution, limiting their range of applicability to problems with explicit functional form for the advection field. In this work we approach the problem of automatically detecting the correct pre-processing transformation in a statistical learning framework by implementing a deep-learning architecture. The purely data-driven method allowed us to generalise the existing approaches of linear subspace manipulation to non-linear hyperbolic problems with unknown advection fields. The proposed algorithm has been validated against simple test cases to benchmark its performances and later successfully applied to a multiphase simulation. {\textcopyright} 2022 Elsevier B.V.</p>},
	keywords = {Advection, Computational complexity, Deep neural network, Deep neural networks, Linear subspace, Multiphase simulations, Non linear, Nonlinear hyperbolic equation, Partial differential equations, Phase space methods, Pre-processing, Principal component analysis, reduced order modeling, Reduced order modelling, Reduced-order model, Shifted-POD},
	doi = {10.1016/j.cma.2022.114687},
	url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85124488633\&doi=10.1016\%2fj.cma.2022.114687\&partnerID=40\&md5=12f82dcaba04c4a7c44f8e5b20101997},
	author = {Davide Papapicco and Nicola Demo and Michele Girfoglio and Giovanni Stabile and Gianluigi Rozza}
}
@booklet {PapapiccoDemoGirfoglioStabileRozza2021,
	title = {The Neural Network shifted-Proper Orthogonal Decomposition: a Machine Learning Approach for Non-linear Reduction of Hyperbolic Equations},
	year = {2021},
	author = {Davide Papapicco and Nicola Demo and Michele Girfoglio and Giovanni Stabile and Gianluigi Rozza}
}
@article {2019,
	title = {A non-intrusive approach for the reconstruction of POD modal coefficients through active subspaces},
	journal = {Comptes Rendus - Mecanique},
	volume = {347},
	year = {2019},
	pages = {873-881},
	abstract = {<p>Reduced order modeling (ROM) provides an efficient framework to compute solutions of parametric problems. Basically, it exploits a set of precomputed high-fidelity solutions{\textemdash}computed for properly chosen parameters, using a full-order model{\textemdash}in order to find the low dimensional space that contains the solution manifold. Using this space, an approximation of the numerical solution for new parameters can be computed in real-time response scenario, thanks to the reduced dimensionality of the problem. In a ROM framework, the most expensive part from the computational viewpoint is the calculation of the numerical solutions using the full-order model. Of course, the number of collected solutions is strictly related to the accuracy of the reduced order model. In this work, we aim at increasing the precision of the model also for few input solutions by coupling the proper orthogonal decomposition with interpolation (PODI){\textemdash}a data-driven reduced order method{\textemdash}with the active subspace (AS) property, an emerging tool for reduction in parameter space. The enhanced ROM results in a reduced number of input solutions to reach the desired accuracy. In this contribution, we present the numerical results obtained by applying this method to a structural problem and in a fluid dynamics one.</p>},
	doi = {https://doi.org/10.1016/j.crme.2019.11.012},
	url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85075379471\&doi=10.1016\%2fj.crme.2019.11.012\&partnerID=40\&md5=dcb27af39dc14dc8c3a4a5f681f7d84b},
	author = {Nicola Demo and Marco Tezzele and Gianluigi Rozza}
}
