Some model reduction techniques for multiple time-scale dynamical systems make use of the identification of low dimensional slow invariant attracting manifolds (SIAM) in order to reduce the dimensionality of the phase space by restriction to the slow flow. The focus of this work is on a proposition and discussion of a …
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48 results for “slow invariant manifolds”
During this last decades, several attempts to construct slow invariant manifold of the Lorenz-Krishnamurthy five-mode model of slow-fast interactions in the atmosphere have been made by various authors. Unfortunately, as in the case of many two-time scales singularly perturbed dynamical systems the various asymptotic p…
The paper studies invariant complex manifolds in holomorphic slow-fast systems.
problem Existence of invariant complex manifolds in holomorphic systems.
method Geometric singular perturbation theory, Fenichel and Briot-Bouquet theories.
result Conditions are provided to guarantee the existence of one-dimensional invariant complex manifolds.
A new geometric method approximates slow invariant manifolds without explicit time-scale separation.
problem Approximating slow invariant manifolds in systems with multiple time-scales.
method Geodesic Stretching and Flow Curvature methods translated into tensorial constructions of Riemannian geometry.
result The method approximates normally attracting invariant manifolds without requiring explicit time-scale separation.
Covariant geometric characterization of slow invariant manifolds: New concepts and viewpointsmath.DS
We point out a new view on slow invariant manifolds (SIM) in dynamical systems which departs from a purely geometric covariant characterization implying coordinate independency. The fundamental idea is to treat the SIM as a well-defined geometric object in phase space and elucidate characterizing geometric properties t…
Efficiently simulates slow dynamics of high-dimensional stochastic systems.
problem Simulating high-dimensional stochastic systems with slow dynamics and fast modes.
method Designs an algorithm to estimate an invariant manifold and its dynamics, averaging out fast modes.
result Efficient simulator of effective dynamics on low-dimensional invariant manifold.
Affine Connection Approach to the Realization of Nonholonomic Constraints by Strong Friction Forcesmath-ph
Study on friction forces for nonholonomic systems using affine connections.
problem Realizing nonholonomic constraints with strong friction forces.
method Affine connection approach, covariant derivatives, recursive procedure.
result Approximations of slip velocities and dynamics up to second order.
The identification of slow invariant manifolds (SIMs) is an essential part in model-order reduction for reactive systems. The mathematical definition of the SIM by Fenichel can be considered unsatisfactory, because it is only applicable to so-called slow-fast system and does not provide the uniqueness of the SIM. Obser…
This work learns effective dynamics from short-term data of stochastic systems.
problem Learning effective dynamics from short-term data of stochastic systems.
method Proposes a novel algorithm using a neural network (Auto-SDE) to learn invariant slow manifold from data.
result Validated through numerical experiments to be accurate, stable, and effective.
Regularization leads to balancedness in deep linear networks.
problem Balancedness in deep linear networks.
method Geometric invariant theory and Riemannian geometry of fibers.
result Balancing flows converge to the balanced manifold at a uniform exponential rate.
We provide a rigorous numerical computation method to validate periodic, homoclinic and heteroclinic orbits as the continuation of singular limit orbits for the fast-slow system with one-dimensional slow variable . Our validation procedure is based on topological tools called isolatin…
Study of manifolds with nonnegative Ricci curvature and slow relative volume growth.
problem Understanding fundamental groups of manifolds with specific volume growth.
method Defined a function RV(s) to describe volume growth and studied fundamental groups with slow relative volume growth.
result If RV(s) grows sublinearly, fundamental groups are almost abelian or finite.
The goal of this paper is to investigate the topological structure of open simply-connected 3-manifolds whose scalar curvature has a slow decay at infinity. In particular, we show that the Whitehead manifold does not admit a complete metric, whose scalar curvature decays slowly, and in fact that any contractible comple…
A framework learns multiscale dynamics from single trajectories using normalizing flows.
problem Learning effective stochastic dynamics from single observed paths of slow variables.
method Data-driven approach based on coupled multiscale SDEs, stochastic averaging, and normalizing flows for density modeling.
result Scalable approach to capturing epistemic uncertainty in multiscale systems.
In this paper, a metric with G holonomy and slow rate of convergence to the cone metric is constructed on a ball inside the cone over the flag manifold.
We show that there are topological obstructions for a noncompact manifold to admit a Riemannian metric with quadratic curvature decay and a volume growth which is slower than that of Euclidean space of the same dimension.
Hierarchical pretraining with slow-fast ODEs
problem Causal self-attention vs. slow-fast ODEs
method Instantiating fast-slow ODE formalism as a concrete neural network
result Equilibrium manifold is exactly the master-equation (ME) stationary distribution
This work interprets SFA through variational inference, relaxing linearity constraints.
problem Recover non-linear SFA from variational inference.
method Probabilistic interpretation of SFA through variational inference, relaxing linearity constraints.
result Reinterprets SFA as a variational framework, allowing slowness as a regularizer to reconstruction loss.
Travel time tomography with adaptive dictionariesphysics.geo-ph
We develop a 2D travel time tomography method which regularizes the inversion by modeling groups of slowness pixels from discrete slowness maps, called patches, as sparse linear combinations of atoms from a dictionary. We propose to use dictionary learning during the inversion to adapt dictionaries to specific slowness…
Method learns dynamics of slow variables from stochastic data.
problem Modeling unknown multiscale stochastic systems with limited data.
method Data-driven approach to learn effective dynamics from bursts of observation data.
result Generative model accurately captures effective dynamics of slow variables.
Gradient method converges locally linearly for overparameterized Gaussian mixtures.
problem Learning Gaussian mixtures under overparameterization.
method Gradient-based method alternating short descent steps and long Polyak steps.
result Gradient method converges locally linearly to minimizers.
We propose Power Slow Feature Analysis, a gradient-based method to extract temporally slow features from a high-dimensional input stream that varies on a faster time-scale, as a variant of Slow Feature Analysis (SFA) that allows end-to-end training of arbitrary differentiable architectures and thereby significantly ext…
Time-lagged autoencoders (TAEs) have been proposed as a deep learning regression-based approach to the discovery of slow modes in dynamical systems. However, a rigorous analysis of nonlinear TAEs remains lacking. In this work, we discuss the capabilities and limitations of TAEs through both theoretical and numerical an…
Study reveals conditions for neural networks to forget learned features.
problem Understanding feature unlearning in neural networks.
method Infinite-width limit analysis with stochastic gradient descent, fast-slow dynamics.
result Conditions for feature unlearning are determined by the strength of nonlinear terms and initial weights.
New methods compute Alexander polynomials for complex knots.
problem Efficiently computing higher order Alexander polynomials for complex knots.
method Developed new algorithms to compute the Smith normal form of Alexander matrices.
result Computed Alexander polynomials for knots up to 100 crossings.
Derives a biologically plausible neural network for Slow Feature Analysis.
problem Learning latent features from time series data.
method Starting from an SFA objective, derives Bio-SFA with a biologically plausible neural network implementation.
result Validates Bio-SFA on naturalistic stimuli, reproducing interesting properties of brain cells.
Extended Predictable Feature Analysis (PFAx) [Richthofer and Wiskott, 2017] is an extension of PFA [Richthofer and Wiskott, 2015] that allows generating a goal-directed control signal of an agent whose dynamics has previously been learned during a training phase in an unsupervised manner. PFAx hardly requires assumptio…
Let N be an irreducible, compact 3-manifold with empty or toroidal boundary which is not a closed graph manifold. Using recent work of Agol, Kahn-Markovic and Przytycki-Wise we will show that pi_1(N) admits a cofinal filtration with `fast' growth of Betti numbers as well as a cofinal filtration of pi_1(N) with `slow' g…
We study the stability of symmetric trajectories of a particle on the Lie group whose motion is governed by an invariant metric and an invariant potential. Our method is to reduce the number of degrees of freedom at {\em singular} values of the momentu…
New findings on magnetic geodesic flows and periodic motions.
problem Characterizing superintegrable systems in magnetic geodesic flows.
method Analyzing rotationally symmetric magnetic geodesic flows.
result All sufficiently slow motions in a central magnetic field are periodic under specific curvature and homogeneity conditions.
Paper introduces slow kill for efficient large-scale variable screening.
problem Challenges in variable selection and parameter estimation for big data.
method Nonconvex constrained optimization, adaptive \(\ell_2\)-shrinkage, and increasing learning rates.
result Slow kill outperforms state-of-the-art algorithms in various situations.
The paper extends hypothesis testing to non-diagonalizable matrices, improving network statistics inference.
problem Testing on non-diagonalizable matrices for network statistics.
method Generalizes Wald and t-tests to non-symmetric matrices, controlling convergence rates.
result Improved inference on network statistics from directed networks.
Minimalistic unsupervised learning with sparse manifold transform achieves SOTA performance.
problem Achieving state-of-the-art unsupervised learning performance without complex engineering.
method Sparse manifold transform, leveraging sparse coding, manifold learning, and slow feature analysis.
result 99.3% KNN top-1 accuracy on MNIST, 81.1% on CIFAR-10, and 53.2% on CIFAR-100.
Tian and Yau constructed a complete Ricci-flat Kähler metric on the complement of an ample and smooth anticanonical divisor. We inquire into the behaviour of this metric towards the boundary divisor and prove a slow decay rate of the difference to an appropriate explicitely given referential metric.
Study the averaging principle for non-autonomous slow-fast systems and apply it to financial local stochastic volatility models.
problem Understanding the behavior of non-autonomous slow-fast systems of stochastic differential equations.
method Prove the averaging principle under specific conditions and apply it to a financial model.
result Prices of derivatives converge to those calculated using the limit model under a risk-neutral measure.
Quasiclassical generalized Weierstrass representation for highly corrugated surfaces with slow modulation in the three-dimensional space is proposed. Integrable deformations of such surfaces are described by the dispersionless Veselov-Novikov hierarchy.
Paper shows how SFA fits into FBM framework for time series separation.
problem Identifying time series decomposition in flow-based models.
method Combining SFA and FBM to make time series decomposition identifiable.
result Time series decomposition becomes identifiable using SFA and FBM.
Slow-Growing Treesstat.ML
A single slow-growing tree matches Random Forest's performance.
problem Matching Random Forest's performance with a single tree.
method SGT uses a learning rate to tame CART's greedy algorithm, improving on greedy ML algorithms.
result SGT and tree ensembles like Booging, BT, and RF improve performance.
Researchers reconstruct stiffness tensors from limited data in anisotropic elasticity.
problem Reconstructing stiffness tensors from partial data around one polarization.
method Using algebraic geometry and slowness surfaces, the approach leverages the algebraic geometry of families of slowness surfaces.
result For tensors in a dense open subset, a small amount of data around one polarization uniquely determines the entire slowness surface and stiffness tensor.
Quantum graphical models (QGMs) extend the classical framework for reasoning about uncertainty by incorporating the quantum mechanical view of probability. Prior work on QGMs has focused on hidden quantum Markov models (HQMMs), which can be formulated using quantum analogues of the sum rule and Bayes rule used in class…
We present a signal representation framework called the sparse manifold transform that combines key ideas from sparse coding, manifold learning, and slow feature analysis. It turns non-linear transformations in the primary sensory signal space into linear interpolations in a representational embedding space while maint…
We introduce two numerical conjugacy invariants for dynamical systems -- the complexity and weak complexity indices -- which are well-suited for the study of "completely integrable" Hamiltonian systems. These invariants can be seen as "slow entropies", they describe the polynomial growth rate of the number of balls (fo…
Slow feature analysis (SFA) is a method for extracting slowly varying driving forces from quickly varying nonstationary time series. We show here that it is possible for SFA to detect a component which is even slower than the driving force itself (e.g. the envelope of a modulated sine wave). It is shown that it depends…
L-CNNs approximate gauge actions, revealing fixed points with no lattice artifacts.
problem Approximating gauge actions with lattice artifacts.
method Lattice gauge-equivariant convolutional neural networks (L-CNNs).
result L-CNNs provide fixed point actions with no lattice artifacts.
Paper separates financial time series into fast and slow components.
problem Multiscale behavior in financial time series data.
method Uses variance and tail stationarity criteria as generalized eigenvalue problems.
result Identifies slow and fast components in asset returns and prices.
New algorithm processes Riemannian data more efficiently.
problem High memory usage and slow speed in previous Riemannian HMMs.
method Online algorithm based on Baum-Welch adapted for Riemannian manifolds.
result Significant improvements in speed and efficiency.
Slow feature analysis (SFA) is an unsupervised-learning algorithm that extracts slowly varying features from a multi-dimensional time series. A supervised extension to SFA for classification and regression is graph-based SFA (GSFA). GSFA is based on the preservation of similarities, which are specified by a graph struc…
Paper shows hard computational limits for invariant causal prediction.
problem Hard computational limits for invariant causal prediction.
method Distributionally robust estimator with ellipse-shaped uncertain set.
result Estimation error rate can be arbitrarily slow for computationally efficient algorithms.