This paper contains a summary of mathematical researches of stochastic properties of the long time behavior of a continuously observed (and interactively controlled) quantum--field top. Applications to interactively controlled stochastic computer-graphic dynamical systems are also discussed.
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48 results for “long-time control”
New control theory shows neural networks can be sparsely active over time.
problem Optimizing neural networks for long-time control with sparsity constraints.
method Proving optimal controls vanish after a positive time and providing a stability estimate.
result Optimal controls for -penalized neural ODEs are sparsely active over time.
In this paper we study Inverse Mean Curvature Flow (IMCF) on manifolds that are conformal to a warped product manifold. To this end, we show how the gradient conformal vector field in warped product manifolds is related to the conformal vector field on the conformal metric and use this to gain control of the flow in or…
Novel method controls complex physical systems over long time frames.
problem Controlling complex nonlinear physical systems over long time frames.
method Hierarchical predictor-corrector scheme with separate planning and control networks.
result Successfully controls complex physical systems like incompressible Navier-Stokes equations.
Gradient flows for knot energies ensure long-term existence of knotted loops.
problem Ensuring long-term existence of knotted loops under various energies.
method Banach gradient flows, curves of maximal slope, logarithmic strain control.
result Established long-time existence of gradient flows for knot energies.
In this paper we analyze the long-time behavior of 3 dimensional Ricci flows with surgery. Our main result is that if the surgeries are performed correctly, then only finitely many surgeries occur and after some time the curvature is bounded by . This result confirms a conjecture of Perelman. In the course of…
Paper introduces mcTangent for real-time dynamical systems.
problem Real-time accurate solutions for complex dynamical systems.
method Synergy of tangent slope learning, model-constrained approach, sequential learning, and data randomization.
result Robust and long-time accurate solutions for various equations.
State space models (SSMs) are a flexible approach to modeling complex time series. However, inference in SSMs is often computationally prohibitive for long time series. Stochastic gradient MCMC (SGMCMC) is a popular method for scalable Bayesian inference for large independent data. Unfortunately when applied to depende…
Study on mean field games with singular controls and their applications.
problem Optimal productivity expansion in dynamic oligopolies.
method Existence and uniqueness of mean field equilibria through nonlinear equations, Abelian limit for discounted and ergodic games.
result Valid connection between discounted and ergodic games, approximation of Nash equilibria.
Neural RDEs extend CDEs to irregular time series.
problem Modeling long irregular time series efficiently.
method Representing time series through log-signature and solving RDEs.
result Significant training speed-ups and improved model performance.
Gradient flossing stabilizes RNN training by controlling Lyapunov exponents.
problem Gradient instability in RNNs leading to exploding and vanishing gradients.
method Regularizing Lyapunov exponents through backpropagation using differentiable linear algebra.
result Gradient flossing improves RNN training success rate and convergence speed.
This survey reviews portfolio selection problem for long-term horizon. We consider two objectives: (i) maximize the probability for outperforming a target growth rate of wealth process (ii) minimize the probability of falling below a target growth rate. We study the asymptotic behavior of these criteria formulated as l…
This is the first of a series of papers on the long-time behavior of 3 dimensional Ricci flows with surgery. In this paper we first fix a notion of Ricci flows with surgery, which will be used in this and the following three papers. Then we review Perelman's long-time estimates and generalize them to the case in which …
Long-time accuracy of ensemble Kalman filters for chaotic and machine-learned dynamical systemsmath.DS
This paper proves long-time accuracy of ensemble Kalman filters for chaotic and machine-learned systems.
problem Ensuring long-term accuracy of ensemble Kalman filters for complex dynamical systems.
method Established conditions for long-time accuracy of ensemble Kalman filters for chaotic and machine-learned dynamical systems.
result Ensemble Kalman filters maintain small estimation error over long time horizons for chaotic and machine-learned systems.
For an immortal Ricci flow on an -dimensional closed manifold, we show the following convergence results: (1) if the curvature and diameter are uniformly bounded, then any unbounded sequence of time slices sub-converges to a Riemannian orbifold; (2) if the flow is type-III with diameter growth controlled …
Prove long-time existence of pluriclosed flow on certain fibrations
problem Long-time existence of pluriclosed flow on fibrations
method General theorem on holomorphic submersions
result Long-time existence of pluriclosed flow on nilmanifolds, almost-abelian solvmanifolds, and certain complex surfaces
New criteria for long-time existence of parabolic flow from 11D supergravity.
problem Establishing long-time existence of a parabolic flow from 11D supergravity.
method Using only Ricci curvatures and their derivatives, along with 4-forms, to establish long-time existence.
result A new criteria for long-time existence of the parabolic flow.
The study provides a criterion for diffeomorphism via long-time Ricci flow.
problem Understanding conditions for diffeomorphism in geometric flows.
method Long-time Ricci flow criterion for diffeomorphism.
result Affirmative answer to manifold diffeomorphism in dimension 4.
Data-driven modelling and synthesis of motion is an active research area with applications that include animation, games, and social robotics. This paper introduces a new class of probabilistic, generative, and controllable motion-data models based on normalising flows. Models of this kind can describe highly complex d…
Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.
problem Existence and convergence of twisted Calabi flow on compact Kähler manifolds.
method Analysis of a family of twisted Calabi flows connecting J-flow and Calabi flow, showing long-time existence and convergence to cscK metrics.
result Long-time existence and convergence of twisted Calabi flow to cscK metrics, implying openness of continuity method.
The anomaly flow on a complex 3-fold is studied with integral Shi-type estimates and long-time existence conditions.
problem Long-time existence of the anomaly flow on a compact complex 3-fold.
method Integral Shi-type estimates adapted from integration-by-parts arguments, with a smallness condition on the slope parameter.
result Long-time existence of the anomaly flow on a compact complex 3-fold under a smallness condition on the slope parameter.
A new RNN model tackles long-time dependencies with fast, invertible, and memory-efficient hidden states.
problem Challenges in processing sequential inputs with long-time dependencies in RNNs.
method A novel RNN architecture based on a Hamiltonian system of oscillators.
result The proposed RNN mitigates exploding and vanishing gradient problems, providing state-of-the-art performance.
Develops a new parabolic equation for surfaces, proving long-time existence and convergence.
problem Extending elliptic equations to parabolic settings for surfaces.
method Introduces a parabolic analogue of the elliptic split-type Monge-Ampère equation.
result Proves long-time existence and convergence conditions for the new equation.
Proves long-time Ricci flow existence and topological rigidity for pinched integral curvature manifolds.
problem Proving long-time existence and topological rigidity for manifolds with pinched scale-invariant integral curvature.
method Proves long-time existence of Ricci flow for manifolds with bounded curvature and pinched scale-invariant integral curvature, converging to a flat metric.
result Flow converges to a flat metric, implying topological rigidity of the manifold.
Researchers prove long-time existence for two landmark Brownian motion.
problem Proving long-time existence of Brownian motion on configurations of two landmarks.
method Classification and analysis of long-time existence for configurations of exactly two landmarks, using a radial kernel.
result For configurations of exactly two landmarks, long-time existence is possible for certain kernels, but not for others.
Flow preserves volume on flat torus, converging to stable set.
problem Volume preservation in discrete mean curvature flow on flat torus.
method Discrete mean curvature flow, quantitative Alexandrov estimate, characterization in 2D.
result Flow converges exponentially fast to stable set.
Efficiently predicts long-time dynamics of quantum spin models using MLP regression.
problem Challenges in calculating long-time expectation values for quantum spin models.
method Utilized a multi-layer perceptron (MLP) model for regression on matrix product states (MPS) expectation values.
result Significantly reduced computational cost for generating long-time dynamics while maintaining high accuracy.
Study long-term asset liquidation behavior with external flows.
problem Investigate optimal liquidation in presence of external flows.
method Convergence analysis of BSDEs for value function and strategy.
result Long-term liquidation may not occur due to external flows.
New algorithm reduces training time for deep learning in financial hedging.
problem Optimal hedging in markets with transaction costs.
method ST-Hedging algorithm combining deep learning and FBSDE solver.
result Achieves state-of-the-art performance and scalability.
Generalized Yamabe Flowsmath.DG
Introduces generalized Yamabe flows with long-time existence and convergence results.
problem Yamabe flow and its limitations.
method Introduces a family of conformal flows generalizing the classical Yamabe flow and proves long-time existence and convergence.
result Long-time existence and convergence for a large class of generalized Yamabe flows.
Paper proposes SCott optimizer to reduce forecasting model training variance.
problem Large variance in gradient estimation for forecasting models.
method Stratified sampling and control variate to reduce gradient variance.
result SCott optimizer converges faster on time series forecasting problems.
Solves long-time solutions for a specific equation on hyperkähler manifolds.
problem Finding solutions to a specific equation on hyperkähler manifolds.
method Introduced a parabolic quaternionic Monge-Ampère equation and proved its long-time solvability.
result Smooth convergence to a solution of the quaternionic Monge-Ampère equation.
We show that three-dimensional homogeneous Ricci flow solutions that admit finite-volume quotients have long-time limits given by expanding solitons. We show that the same is true for a large class of four-dimensional homogeneous solutions. We give an extension of Hamilton's compactness theorem that does not assume a l…
The paper explores new risk models for autonomous driving.
problem Risk management and actuarial modeling for autonomous vehicles.
method Examines technical difficulties and proposes a novel risk model.
result The new model better reflects real-world driving safety.
Study shows uniform bounds on torsion and curvature for Chern-Ricci flow solutions.
problem Understanding singularity types in long-time solutions of Chern-Ricci flow.
method Extended results from Kähler-Ricci flow to Chern-Ricci flow, focusing on uniform bounds on torsion and curvature.
result Uniform bounds on torsion and curvature for solutions starting from metrics of the same class.
Study models Ricci flow on complex surfaces, showing mixed behavior.
problem Understanding long-time behavior of Ricci flow on complex surfaces.
method BiLipschitz models for 4-manifolds (minimal surfaces of general type).
result Exhibits a combination of expanding and static behavior.
Study shows long-term flow on special manifolds with positive Yamabe constant.
problem Analyzing long-time behavior of Yamabe flow on singular spaces.
method Formulated axioms for long-time existence, used parabolic Moser iteration for bounds.
result Established long-time existence of normalized Yamabe flow on specified manifolds.
We recast the Calabi flow in DeGiorgi's language of minimizing movements. We establish the long time existence of minimizing movements for K-energy with arbitrary initial condition. Furthermore we establish some a priori regularity of these solutions, and that sufficiently regular minimizing movements are smooth soluti…
Turnpike property applies to optimal control of PDEs and ResNets.
problem Optimal control of PDEs and ResNets.
method Mathematical formalization and controllability analysis.
result Optimal controls and states are nearly constant over most of the time.
Study the long-time behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.
problem Understanding the long-time behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.
method Monotonicity of eigenvalues of mean curvature, convergence to geometric invariants.
result Eigenvalues of mean curvature converge to geometric invariants in the Gauduchon case.
Paper proves curvature estimates for a specific flow on Kähler manifolds.
problem Proving local curvature estimates for a specific flow on Kähler manifolds.
method Proves local curvature estimates for the -LYZ flow over Kähler manifolds.
result Generalizes the long time existence of the flow.
Kahler-Ricci flow long-time behavior and initial data
problem Relationship between Kahler-Ricci flow and initial data
method Investigate long-time behavior
result Asymptotic profiles and non-trivial breathers
Existence of balanced embedding proved for complex manifold into infinite-dimensional space.
problem Balanced embedding of non-compact complex manifolds into infinite-dimensional projective space.
method Gradient flow in a Hilbert space, long-time existence established by perturbation, convergence depends on a priori bounds.
result Existence of balanced embedding proved in a model case.
We prove long-time existence and convergence results for spacelike solutions to mean curvature flow in the pseudo-Euclidean space , which are entire or defined on bounded domains and satisfying Neumann or Dirichlet boundary conditions. As an application, we prove long-time existence and convergence of…
Study shows Ricci flow's convergence and harmonic map heat flow's long-time existence.
problem Analyzing convergence of Ricci flow and harmonic map heat flow.
method Established long-time existence of harmonic map heat flow between Ricci flow and shrinker.
result Ricci flow converges exponentially to compact integrable shrinkers and at singularities modelled on the shrinker.
We prove that at a finite singular time for the Harmonic Ricci Flow on a surface of positive genus both the energy density of the map component and the curvature of the domain manifold have to blow up simultaneously. As an immediate consequence, we obtain smooth long-time existence for the Harmonic Ricci Flow with larg…
We study the long time behaviour of Ricci flow with bubbling-off on a possibly noncompact -manifold of finite volume whose universal cover has bounded geometry. As an application, we give a Ricci flow proof of Thurston's hyperbolisation theorem for -manifolds with toral boundary that generalizes Perelman's proof …
In this paper we study the long time existence of the Ricci-harmonic flow in terms of scalar curvature and Weyl tensor which extends Cao's result \cite{Cao2011} in the Ricci flow. In dimension four, we also study the integral bound of the "Riemann curvature" for the Ricci-harmonic flow generalizing a recently result of…