Manifold curvature and Ehrenfest forces with a moving basis

Known force terms arising in the Ehrenfest dynamics of quantum electrons and classical nuclei, due to a moving basis set for the former, can be understood in terms of the curvature of the manifold hosting the quantum states of the electronic subsystem. Namely, the velocity-dependent terms appearing in the Ehrenfest forces on the nuclei acquire a geometrical meaning in terms of the intrinsic curvature of the manifold, while Pulay terms relate to its extrinsic curvature.

An airfoil geometric-feature extraction and discrepant data fusion learning method

The perception of geometric-features of airfoils is the basis in aerodynamic area for performance prediction, parameterization, aircraft inverse design, etc. There are three approaches to percept the geometric shape of airfoils, namely manual design of airfoil geometry parameters, polynomial definition and deep learning. The first two methods directly define geometric-features or polynomials of airfoil curves, but the number of extracted features is limited. Deep learning algorithms can extract a large number of potential features (called latent features). However, the features extracted by deep learning lack explicit geometrical meaning. Motivated by the advantages of polynomial definition and deep learning, we propose a geometric-feature extraction method (named Bézier-based feature extraction, BFE) for airfoils, which consists of two parts: manifold metric feature extraction and geometric-feature fusion encoder (GF encoder). Manifold metric feature extraction, with the help of the Bézier curve, captures manifold metrics (a sort of geometric-features) from tangent space of airfoil curves, and the GF-encoder combines airfoil coordinate data and manifold metrics together to form novel fused geometric-features. To validate the feasibility of the fused geometric-features, two experiments based on the public UIUC airfoil dataset are conducted. Experiment I is used to extract manifold metrics of airfoils and export the fused geometric-features. Experiment II, based on the Multi-task learning (MTL), is used to fuse the discrepant data (i.e., the fused geometric-features and the flight conditions) to predict the aerodynamic performance of airfoils. The results show that the BFE can generate more smooth and realistic airfoils than Auto-Encoder, and the fused geometric-features extracted by BFE can be used to reduce the prediction errors of C L and C D .

A Geometry-Based Deep Learning Feature Extraction Scheme for Airfoils

The perception of geometry-features of airfoils is the basis in aerodynamic area for performance prediction, parameterization, aircraft inverse design, etc. There are three approaches to percept the geometric shape of an airfoil, namely manual design of airfoil geometry parameter, polynomial definition and deep learning. The first two methods can directly extract geometry-features of airfoils or polynomial equations of airfoil curves, but the number of features extracted is limited. While deep learning algorithms can extract a large number of potential features (called latent features), however, the features extracted by deep learning are lacking of explicit geometrical meaning. Motivated by the advantages of polynomial definition and deep learning, we propose a geometry-based deep learning feature extraction scheme (named Bézier-based feature extraction, BFE) for airfoils, which consists of two parts: manifold metric feature extraction and geometry-feature fusion encoder (GF encoder). Manifold metric feature extraction, with the help of the Bézier curve, captures features from tangent space of airfoil curves, and GF encoder combines airfoil coordinate data and manifold metrics together to form a novel feature representation. A public UIUC airfoil dataset is used to verify the proposed BFE. Compared with classic Auto-Encoder, the mean square error (MSE) of BFE is reduced by 17.97% ~29.14%.

The Geometrical Meaning of Spinors Lights the Way to Make Sense of Quantum Mechanics

This paper aims at explaining that a key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R3. In the three-dimensional rotation group, the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding that is acquired to derive the free-space Dirac equation from scratch, proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine’s statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further, we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.

Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations

A geometrical formulation for adjoint-symmetries as one-forms is studied for general partial differential equations (PDEs), which provides a dual counterpart of the geometrical meaning of symmetries as tangent vector fields on the solution space of a PDE. Two applications of this formulation are presented. Additionally, for systems of evolution equations, adjoint-symmetries are shown to have another geometrical formulation given by one-forms that are invariant under the flow generated by the system on the solution space. This result is generalized to systems of evolution equations with spatial constraints, where adjoint-symmetry one-forms are shown to be invariant up to a functional multiplier of a normal one-form associated with the constraint equations. All of the results are applicable to the PDE systems of interest in applied mathematics and mathematical physics.

Numerical Non-Linear Modelling Algorithm Using Radial Kernels on Local Mesh Support

Estimation problems are frequent in several fields such as engineering, economics, and physics, etc. Linear and non-linear regression are powerful techniques based on optimizing an error defined over a dataset. Although they have a strong theoretical background, the need of supposing an analytical expression sometimes makes them impractical. Consequently, a group of other approaches and methodologies are available, from neural networks to random forest, etc. This work presents a new methodology to increase the number of available numerical techniques and corresponds to a natural evolution of the previous algorithms for regression based on finite elements developed by the authors improving the computational behavior and allowing the study of problems with a greater number of points. It possesses an interesting characteristic: Its direct and clear geometrical meaning. The modelling problem is presented from the point of view of the statistical analysis of the data noise considered as a random field. The goodness of fit of the generated models has been tested and compared with some other methodologies validating the results with some experimental campaigns obtained from bibliography in the engineering field, showing good approximation. In addition, a small variation on the data estimation algorithm allows studying overfitting in a model, that it is a problematic fact when numerical methods are used to model experimental values.