Cost function for low-dimensional manifold topology assessment

Abstract

In reduced-order modeling, complex systems that exhibit high state-space dimensionality are described and evolved using a small number of parameters. These parameters can be obtained in a data-driven way, where a high-dimensional dataset is projected onto a lower-dimensional basis. A complex system is then restricted to states on a low-dimensional manifold where it can be efficiently modeled. While this approach brings computational benefits, obtaining a good quality of the manifold topology becomes a crucial aspect when models, such as nonlinear regression, are built on top of the manifold. Here, we present a quantitative metric for characterizing manifold topologies. Our metric pays attention to non-uniqueness and spatial gradients in physical quantities of interest, and can be applied to manifolds of arbitrary dimensionality. Using the metric as a cost function in optimization algorithms, we show that optimized low-dimensional projections can be found. We delineate a few applications of the cost function to datasets representing argon plasma, reacting flows and atmospheric pollutant dispersion. We demonstrate how the cost function can assess various dimensionality reduction and manifold learning techniques as well as data preprocessing strategies in their capacity to yield quality low-dimensional projections. We show that improved manifold topologies can facilitate building nonlinear regression models.

Local manifold learning and its link to domain-based physics knowledge

“…They can be further exploited in the future to provide localized closure models for high-fidelity combustion simulations, or to improve local kinetic schemes. Future research can also focus on quantifying the parameterization quality coming from local PCA, for example using recently proposed quantitative metrics [47,48]. We note that VQPCA can be an adequate clustering technique whenever linear subspaces are anticipated in the data, regardless of the origin of the data.…”

Data-driven models and digital twins for sustainable combustion technologies

“…A quantitative manifold-informed method to assess the manifold topology has been proposed, 85 opening the way to an optimal (from the topological perspective) parameterization of the low-dimensional manifold using both linear (e.g., using PCA) and non-linear (e.g., autoencoders) projections, and the adaptive mapping/regression of the variables of interest accounting for local manifold characteristics. Using linear/non-linear activation functions (for the encoding and decoding processes, respectively) combined with topology-aware loss functions is particularly appealing.…”

Reduced-order modeling of supersonic fuel–air mixing in a multi-strut injection scramjet engine using machine learning techniques

“…In order to determine the best scaling for the input parameters, we use a recently proposed cost function, , that can help assess the quality of data parameterization for regression tasks [45,53]. By minimizing the cost value, we determine which scaling factor shall be applied on the design variables (independent variables) to yield good regressibility of each objective function (dependent variables).…”