Enhanced parametric shape descriptions in PGD-based space separated representations

Space separation within the Proper Generalized Decomposition—PGD—rationale allows solving high dimensional problems as a sequence of lower dimensional ones. In our former works, different geometrical transformations were proposed for addressing complex shapes and spatially non-separable domains. Efficient implementation of separated representations needs expressing the domain as a product of characteristic functions involving the different space coordinates. In the case of complex shapes, more sophisticated geometrical transformations are needed to map the complex physical domain into a regular one where computations are performed. This paper aims at proposing a very efficient route for accomplishing such space separation. A NURBS-based geometry representation, usual in computer aided design—CAD—, is retained and combined with a fully separated representation for allying efficiency (ensured by the fully separated representations) and generality (by addressing complex geometries). Some numerical examples are considered to prove the potential of the proposed methodology.

Using top-down gating to optimally balance shared versus separated task representations

Human adaptive behavior requires continually learning and performing a wide variety of tasks, often with very little practice. To accomplish this, it is crucial to separate neural representations of different tasks in order to avoid interference. At the same time, sharing neural representations supports generalization and allows faster learning. Therefore, a crucial challenge is to find an optimal balance between shared versus separated representations. Typically, models of human cognition employ top-down gating signals to separate task representations, but there exist surprisingly little systematic computational investigations of how such gating is best implemented. We identify and systematically evaluate two crucial features of gating signals. First, top-down input can be processed in an additive or multiplicative manner. Second, the gating signals can be adaptive (learned) or non-adaptive (random). We cross these two features, resulting in four gating models which are tested on a variety of input datasets and tasks with different degrees of stimulus-action mapping overlap. The multiplicative adaptive gating model outperforms all other models in terms of accuracy. Moreover, this model develops hidden units that optimally share representations between tasks. Specifically, different than the binary approach of currently popular latent state models, it exploits partial overlap between tasks.

Fast Computation of Multi-Parametric Electromagnetic Fields in Synchronous Machines by Using PGD-Based Fully Separated Representations

A novel Model Order Reduction (MOR) technique is developed to compute high-dimensional parametric solutions for electromagnetic fields in synchronous machines. Specifically, the intrusive version of the Proper Generalized Decomposition (PGD) is employed to simulate a Permanent-Magnet Synchronous Motor (PMSM). The result is a virtual chart allowing real-time evaluation of the magnetic vector potential as a function of the operation point of the motor, or even as a function of constructive parameters, such as the remanent flux in permanent magnets. Currently, these solutions are highly demanded by the industry, especially with the recent developments in the Electric Vehicle (EV). In this framework, standard discretization techniques require highly time-consuming simulations when analyzing, for instance, the noise and vibration in electric motors. The proposed approach is able to construct a virtual chart within a few minutes of off-line simulation, thanks to the use of a fully separated representation in which the solution is written from a series of functions of the space and parameters coordinates, with full space separation made possible by the use of an adapted geometrical mapping. Finally, excellent performances are reported when comparing the reduced-order model with the more standard and computationally costly Finite Element solutions.

On the Rational Boundedness of Cognitive Control: Shared Versus Separated Representations

One of the most fundamental and striking limitations of human cognition appears to be a constraint in the number of control-dependent processes that can be executed at one time. This constraint motivates one of the most influential tenets of cognitive psychology: that cognitive control relies on a central, limited capacity processing mechanism that imposes a seriality constraint on processing. Here we provide a formally explicit challenge to this view. We argue that the causality is reversed: the constraints on control-dependent behavior reflect a rational bound that control mechanisms impose on processing, to prevent processing interference that arises if two or more tasks engage the same resource to be executed. We use both mathematical and numerical analyses of shared representations in neural network architectures to articulate the theory, and demonstrate its ability to explain a wide range of phenomena associated with control-dependent behavior. Furthermore, we argue that the need for control, arising from the shared use of the same resources by different tasks, reflects the optimization of a fundamental tradeoff intrinsic to network architectures: the increase in learning efficacy associated with the use of shared representations, versus the efficiency of parallel processing (i.e., multitasking) associated with task-dedicated representations. The theory helps frame a formally rigorous, normative approach to the tradeoff between control-dependent processing versus automaticity, and relates to a number of other fundamental principles and phenomena concerning cognitive function, and computation more generally.

Analytical solution for the stress field of hierarchical defects: multiscale framework and applications

Hierarchical defects are defined as adjacent defects at different length scales. Involved are the two scales where the stress field distribution is interrelated. Based on the complex variable method and conformal mapping, a multiscale framework for solving the problems of hierarchical defects is formulated. The separated representations of mapping function, the governing equations of potentials, and the stress field are subsequently obtained. The proposed multiscale framework can be used to solve a variety of simplified engineering problems. The case in point is the analytical solution of a macroscopic elliptic hole with a microscopic circular edge defect. The results indicate that the microscopic defect aggregates the stress concentration on the macroscopic defect and likely leads to global propagation and rupture. Multiple micro-defects have interactive effects on the distribution of the stress field. The level of stress concentration may be reduced by the coalescence of micro-defects. This work provides a unified method to analytically investigate the influence of edge micro-defects within the scope of multiscale hierarchy. The formulated multiscale approach can also be potentially applied to materials with hierarchical defects, such as additive manufacturing and bio-inspired materials.