Grassmannian diffusion maps based surrogate modeling via geometric harmonics

In this paper, a novel surrogate model based on the Grassmannian diffusion maps (GDMaps) and utilizing geometric harmonics is developed for predicting the response of engineering systems and complex physical phenomena. The method utilizes the GDMaps to obtain a low-dimensional representation of the underlying behavior of physical/mathematical systems with respect to uncertainties in the input parameters. Using this representation, geometric harmonics, an out-of-sample function extension technique, is employed to create a global map from the space of input parameters to a Grassmannian diffusion manifold. Geometric harmonics is also employed to locally map points on the diffusion manifold onto the tangent space of a Grassmann manifold. The exponential map is then used to project the points in the tangent space onto the Grassmann manifold, where reconstruction of the full solution is performed. The performance of the proposed surrogate modeling is verified with three examples. The first problem is a toy example used to illustrate the development of the technique. In the second example, errors associated with the various mappings employed in the technique are assessed by studying response predictions of the electric potential of a dielectric cylinder in a homogeneous electric field. The last example applies the method for uncertainty prediction in the strain field evolution in a model amorphous material using the shear transformation zone (STZ) theory of plasticity. In all examples, accurate predictions are obtained, showing that the present technique is a strong candidate for the application of uncertainty quantification in large-scale models.