33,552 research outputs found
Forward-backward equations for nonlinear propagation in axially-invariant optical systems
We present a novel general framework to deal with forward and backward
components of the electromagnetic field in axially-invariant nonlinear optical
systems, which include those having any type of linear or nonlinear transverse
inhomogeneities. With a minimum amount of approximations, we obtain a system of
two first-order equations for forward and backward components explicitly
showing the nonlinear couplings among them. The modal approach used allows for
an effective reduction of the dimensionality of the original problem from 3+1
(three spatial dimensions plus one time dimension) to 1+1 (one spatial
dimension plus one frequency dimension). The new equations can be written in a
spinor Dirac-like form, out of which conserved quantities can be calculated in
an elegant manner. Finally, these new equations inherently incorporate
spatio-temporal couplings, so that they can be easily particularized to deal
with purely temporal or purely spatial effects. Nonlinear forward pulse
propagation and non-paraxial evolution of spatial structures are analyzed as
examples.Comment: 11 page
Simultaneous Learning of Nonlinear Manifold and Dynamical Models for High-dimensional Time Series
The goal of this work is to learn a parsimonious and informative representation for high-dimensional time series. Conceptually, this comprises two distinct yet tightly coupled tasks: learning a low-dimensional manifold and modeling the dynamical process. These two tasks have a complementary relationship as the temporal constraints provide valuable neighborhood information for dimensionality reduction and conversely, the low-dimensional space allows dynamics to be learnt efficiently. Solving these two tasks simultaneously allows important information to be exchanged mutually. If nonlinear models are required to capture the rich complexity of time series, then the learning problem becomes harder as the nonlinearities in both tasks are coupled. The proposed solution approximates the nonlinear manifold and dynamics using piecewise linear models. The interactions among the linear models are captured in a graphical model. By exploiting the model structure, efficient inference and learning algorithms are obtained without oversimplifying the model of the underlying dynamical process. Evaluation of the proposed framework with competing approaches is conducted in three sets of experiments: dimensionality reduction and reconstruction using synthetic time series, video synthesis using a dynamic texture database, and human motion synthesis, classification and tracking on a benchmark data set. In all experiments, the proposed approach provides superior performance.National Science Foundation (IIS 0308213, IIS 0329009, CNS 0202067
Model order reduction for stochastic dynamical systems with continuous symmetries
Stochastic dynamical systems with continuous symmetries arise commonly in
nature and often give rise to coherent spatio-temporal patterns. However,
because of their random locations, these patterns are not well captured by
current order reduction techniques and a large number of modes is typically
necessary for an accurate solution. In this work, we introduce a new
methodology for efficient order reduction of such systems by combining (i) the
method of slices, a symmetry reduction tool, with (ii) any standard order
reduction technique, resulting in efficient mixed symmetry-dimensionality
reduction schemes. In particular, using the Dynamically Orthogonal (DO)
equations in the second step, we obtain a novel nonlinear Symmetry-reduced
Dynamically Orthogonal (SDO) scheme. We demonstrate the performance of the SDO
scheme on stochastic solutions of the 1D Korteweg-de Vries and 2D Navier-Stokes
equations.Comment: Minor revision
Do muscle synergies reduce the dimensionality of behavior?
The muscle synergy hypothesis is an archetype of the notion of Dimensionality Reduction (DR) occurring in the central nervous system due to modular organisation. Towards validating this hypothesis, it is however important to understand if muscle synergies can reduce the state-space dimensionality while suitably achieving task control. In this paper we present a scheme for investigating this reduction, utilising the temporal muscle synergy formulation. Our approach is based on the observation that constraining the control input to a weighted combination of temporal muscle synergies instead constrains the dynamic behaviour of a system in trajectory-specific manner. We compute this constrained reformulation of system dynamics and then use the method of system balancing for quantifying the DR; we term this approach as Trajectory Specific Dimensionality Analysis (TSDA). We then use this method to investigate the consequence of minimisation of this dimensionality for a given task. These methods are tested in simulation on a linear (tethered mass) and a nonlinear (compliant kinematic chain) system; dimensionality of various reaching trajectories is compared when using idealised temporal synergies. We show that as a consequence of this Minimum Dimensional Control (MDC) model, smooth straight-line Cartesian trajectories with bell-shaped velocity profiles are obtained as the solution to reaching tasks in both of the test systems. We also investigate the effect on dimensionality due to adding via-points to a trajectory. The results indicate that a synergy basis and trajectory-specific DR of motor behaviours results from usage of muscle synergy control. The implications of these results for the synergy hypothesis, optimal motor control, developmental skill acquisition and robotics are then discussed
Physics-aware registration based auto-encoder for convection dominated PDEs
We design a physics-aware auto-encoder to specifically reduce the
dimensionality of solutions arising from convection-dominated nonlinear
physical systems. Although existing nonlinear manifold learning methods seem to
be compelling tools to reduce the dimensionality of data characterized by a
large Kolmogorov n-width, they typically lack a straightforward mapping from
the latent space to the high-dimensional physical space. Moreover, the realized
latent variables are often hard to interpret. Therefore, many of these methods
are often dismissed in the reduced order modeling of dynamical systems governed
by the partial differential equations (PDEs). Accordingly, we propose an
auto-encoder type nonlinear dimensionality reduction algorithm. The
unsupervised learning problem trains a diffeomorphic spatio-temporal grid, that
registers the output sequence of the PDEs on a non-uniform
parameter/time-varying grid, such that the Kolmogorov n-width of the mapped
data on the learned grid is minimized. We demonstrate the efficacy and
interpretability of our approach to separate convection/advection from
diffusion/scaling on various manufactured and physical systems.Comment: 10 pages, 6 figure
Kinematics and Kinetics of Capacitated and Non-Capacitated Mouse Sperm
The differences in kinetic measurements of force, work, power, and torque are quantitativly observed between non-capacitated and capacitated sperm. Isomap, a nonlinear dimensionality reduction technique, is used to get higher resolution imaging data of the sperm videos. This allowed accurate calculations of spatial and temporal derivatives to calculate the kinetics of the flagella. The results showed no statistical significance between the kinetic measurements of non- capacitated and capacitated mouse sperm
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