Abstract
Domain Knowledge in Data Analysis: A Geometrical Method for Low-Dimensional Representations of Simulations
Jochen Garcke
Universität Bonn and Fraunhofer SCAI
We first highlight examples for the structured integration of domain knowledge into machine learning in engineering applications. In particular, we discuss grey box machine learning approaches in virtual product development and the interplay of data analysis with modelling and simulation.
We then propose a new data analysis approach where simulations of an industrial product are contained in the space of surface meshes embedded in \( \mathbb{R}^3 \), and assume that distance-preserving transformations exist, albeit unknown, which map simulation to simulation. In this setting, a discrete Laplace–Beltrami operator can be constructed on the mesh, which is invariant under isometric transformations and therefore valid for all simulations.
The eigenfunctions of such an operator are used as a common basis for all (isometric) simulations, and one can use the projection coefficients instead of the full simulations for further analysis. To extend the idea of invariance, we employ a discrete Fokker–Planck operator that, in the continuous limit, converges to an operator invariant under a nonlinear transformation, and use its eigendecomposition accordingly.
The data analysis approach is applied to time-dependent datasets from numerical car crash simulations. One observes that only a few spectral coefficients are necessary to describe the data variability, and low-dimensional structures are obtained. The eigenvectors are seen to recover different independent variation modes such as translation, rotation, or global and local deformations. An effective analysis of the data from bundles of numerical simulations is made possible, in particular an analysis for many simulations over time.
We then propose a new data analysis approach where simulations of an industrial product are contained in the space of surface meshes embedded in \( \mathbb{R}^3 \), and assume that distance-preserving transformations exist, albeit unknown, which map simulation to simulation. In this setting, a discrete Laplace–Beltrami operator can be constructed on the mesh, which is invariant under isometric transformations and therefore valid for all simulations.
The eigenfunctions of such an operator are used as a common basis for all (isometric) simulations, and one can use the projection coefficients instead of the full simulations for further analysis. To extend the idea of invariance, we employ a discrete Fokker–Planck operator that, in the continuous limit, converges to an operator invariant under a nonlinear transformation, and use its eigendecomposition accordingly.
The data analysis approach is applied to time-dependent datasets from numerical car crash simulations. One observes that only a few spectral coefficients are necessary to describe the data variability, and low-dimensional structures are obtained. The eigenvectors are seen to recover different independent variation modes such as translation, rotation, or global and local deformations. An effective analysis of the data from bundles of numerical simulations is made possible, in particular an analysis for many simulations over time.