Study shows limits of certain normalizing flows in higher dimensions.
problem Understanding the representation power of normalizing flows in different dimensions.
method Rigorously established bounds on expressive power of basic normalizing flows.
result Limited representation power in higher dimensions, especially with moderate depth.
Ancient flows by curvature powers in 2D have finite entropy.
problem Existence of non-homothetic ancient flows by powers of curvature in R2. method Determined Morse indices and kernels of the linearized operator of shrinkers. Constructed flows using unstable eigenfunctions.
result Existence of ancient flows with finite entropy.
We present a unified method, based on convex optimization, for managing the power produced and consumed by a network of devices over time. We start with the simple setting of optimizing power flows in a static network, and then proceed to the case of optimizing dynamic power flows, i.e., power flows that change with ti…
The study finds that only round spheres shrink self-similarly under certain curvature flows.
problem Investigating self-similar solutions to curvature flows by high powers of curvature.
method Analyzing closed strictly convex hypersurfaces in Rn+1 under specific curvature flows. result Only round spheres shrink self-similarly under the studied curvature flows.
Study finds solutions to flows by negative curvature powers.
problem Curvature flows with negative powers.
method Closed self-similar solutions in warped product manifolds, proving non-strict convexity.
result Proves self-similar solutions are slices of warped product manifolds.
The paper proves entropy power properties on Riemannian manifolds and Ricci flows.
problem Entropy power on Riemannian manifolds and Ricci flows.
method Proving concavity and convexity of Shannon entropy power for heat and conjugate heat equations on Riemannian manifolds and Ricci flows.
result Entropy power rigidity models on Einstein or quasi Einstein manifolds and shrinking Ricci solitons.
New convex ancient solutions found for flows by high powers of curvature.
problem Existence of closed convex ancient solutions to curvature flows.
method Proves existence of closed convex ancient solutions with specific curvature flow speeds.
result Existence of non-homothetic convex ancient solutions for flows by high powers of curvature.
This paper develops an ensemble learning-based linearization approach for power flow, which differs from the network-parameter based direct current (DC) power flow or other extended versions of linearization. As a novel data-driven linearization through data mining, it firstly applies the polynomial regression (PR) as …
The paper classifies flows of ancient curves in 2D space.
problem Classifying closed convex flows by curvature powers.
method Sub-affine-critical powers of curvature for flow classification.
result Ancient flows converge exponentially to smooth shrinkers.
In this paper, we consider the contracting curvature flow of smooth closed surfaces in 3-dimensional hyperbolic space and in 3-dimensional sphere. In the hyperbolic case, we show that if the initial surface M0 has positive scalar curvature, then along the flow by a positive power α of the mean curvature H, t…
New curves defined by curvature powers studied for variational properties.
problem Characterizing translating solitons in curve flows.
method Variational characterization of generalized elastic curves.
result New variational characterization of grim reaper curve.
In this paper, we study the power of Gaussian curvature flow of a compact convex hypersurface and establish its Harnack inequality when the power is negative. In the Harnack inequality, we require that the absolute value of the power is strictly positive and strictly less than the inverse of the dimension of the hypers…
The study finds complete translating solitons for certain powers of Gaussian curvature in Riemannian products.
problem Exploring translating solitons in Riemannian products with powers of Gaussian curvature.
method Investigating Kα-flows in Riemannian products MimesR for M=Rn,Sn,HFm. result Existence of complete rotational translating solitons for certain values of α in MimesR. This paper proves the existence of self-expanders for a specific curvature flow in Minkowski space.
problem Proving the existence of self-expanders for power of σk curvature flow in Minkowski space.
method Analyzing entire, spacelike, convex hypersurfaces with bounded principal curvatures and applying the σk power curvature flow.
result The flow converges to a convex self-expander satisfying σk(κ[tilde{M}])=(-<X0, ν0>)^α.
We show non-collapsing for the evolution of nearly spherical closed convex curves in \mathbb{R}^2 under power curvature flow using two-point-methods.
This paper aims to systematically and comprehensively initiate a foundation for using concepts from computational differential geometry as instruments for power flow computing and research. At this point we focus our discussion on the static case, with power flow equations given by quadratic functions defined on voltag…
Improves normalizing flows by incorporating data dependencies.
problem Current normalizing flow learning assumes independent data, leading to errors.
method Proposes a likelihood objective with dependencies and efficient learning algorithm.
result Improves density estimation and data generation on real-world data.
Normalizing flows simplify complex distributions through bijective transformations.
problem Defining expressive probability distributions efficiently.
method Bijective transformations on a base distribution.
result Unified perspective on normalizing flows for modeling and inference.
A robust model handles up to 25% of outliers in time-series data for power flow calculations.
problem Handling outliers in time-series data for accurate power flow calculations.
method Robust data-driven process model with Schweppe-type generalized maximum likelihood estimator and projection statistics for outlier weighting.
result The model can handle up to 25% of outliers in the training data set.
Classifies surfaces translating under specific curvature flows.
problem Classifying surfaces translating under flows by sub-affine-critical powers of Gauss curvature.
method Analyzes entire graphs of surfaces translating under flows by sub-affine-critical powers of the Gauss curvature.
result Lists all translating solitons possibly model Type II singularities for convex closed solutions in all positive powers.
The paper studies how convex hypersurfaces evolve under curvature flows in space forms.
problem Understanding the evolution of convex hypersurfaces under curvature flows in different space forms.
method Flow by powers of the Gauss curvature in space forms.
result Convex hypersurfaces under the flow by powers of the Gauss curvature in space forms contract to a point in finite time or converge to geodesic spheres.
We consider flows with normal velocities equal to powers strictly larger than one of the Gauss curvature. Under such flows closed strictly convex surfaces converge to points. In his work on the square of the norm of the second fundamental form, Schnürer proposes criteria for selecting quantities that are suitable for p…
Paper introduces normalizing flows for accurate probabilistic energy forecasting.
problem Uncertainty in renewable energy forecasting for power systems.
method Normalizing flows for direct learning of multivariate stochastic distributions.
result Normalizing flows outperform other deep learning models in probabilistic forecasting.
Flow adjusts curvature to avoid a fixed region, proving bounds and regularity.
problem Adjusting curvature flow to avoid a fixed region.
method Flow by powers of Gauss curvature, proving optimal curvature bounds and regularity.
result Proves optimal curvature bounds and long time existence for all dimensions and powers.
Study explores how scalar functionals evolve under Ricci flow.
problem Understanding the evolution of functionals involving scalar quantities under Ricci flow.
method Deriving explicit expressions for the time derivative of integrals of scalar functionals under extended Ricci flow.
result Explicit expressions for the time derivative of integrals involving scalar functionals under Ricci flow.
GP CC-OPF solves uncertain power grid optimization with Gaussian Process.
problem Uncertainty in power grid operations due to high renewables integration.
method Data-driven Gaussian Process regression for solving non-convex CC-OPF problem.
result Effective economic dispatch optimization in uncertain power grids.
New flow expands hypersurfaces in hyperbolic space, showing round limiting shape for certain powers.
problem Understanding the limiting shape of hypersurfaces expanding in hyperbolic space.
method Introduced shifted inverse curvature flow with positive power p for a smooth curvature function. result For 0<p≤1, limiting shape is always round as maximal existence time is approached. Physics-guided neural network improves power flow analysis.
problem Infeasibility of traditional numerical approaches due to outdated or unavailable PF equations.
method Proposes a physics-guided neural network to learn PF mappings from historical data while constraining by physical laws.
result Physics-guided neural network achieves better performance and generalizability than unconstrained data-driven approaches.
A new method uses Gaussian Processes to solve power flow problems with uncertain renewable and load inputs.
problem Solving power flow problems with uncertain renewable and load inputs.
method Non-parametric Bayesian inference-based uncertainty propagation using Gaussian Processes.
result The method provides reasonably accurate solutions with fewer samples and time compared to Monte-Carlo simulations.
We prove that convex hypersurfaces in Rn+1 contracting under the flow by any power α>n+21 of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the ini…
Optimizes power systems with energy storage under uncertainty using scenario-based method.
problem Optimizing power systems with energy storage, intermittent renewable generation, and uncontrollable loads under uncertainty.
method Developed a novel solution method based on scenario optimization and strategic sampling to solve the chance-constrained optimal power system operation problem.
result The strategic sampling method significantly improves computational efficiency and data-driven convex approximation of power flow.
The paper constructs hypersurfaces translating under powers of Gauss curvature.
problem Existence of hypersurfaces translating under powers of Gauss curvature.
method Constructs complete convex hypersurfaces in R^(n+1) translating under flow by powers of Gauss curvature.
result Existence of translators whose level set converges to various shapes like sphere, simplex, and hypercube.
Study shows how curved surfaces evolve smoothly to spherical shapes.
problem Evolution of curved surfaces with capillary boundaries.
method Volume-preserving curvature flow with power mean curvature speed.
result Convex initial hypersurfaces evolve to spherical caps over time.
We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the m-th mean curvature plus a volume preserving term, including the case of powers of the mean curvature or of the Gauss curvature. We prove that if the initial hypersurface satisfies a …
The paper extends entropy formulas to super Ricci flows on metric measure spaces.
problem Entropy formulas for super Ricci flows on metric measure spaces.
method Extending Perelman's W-entropy and Shannon entropy power to super Ricci flows. result Equivalence between volume non-local collapsing property and lower boundedness of W-entropy on RCD(0,N) spaces. New framework explains normalizing flows' power and limitations.
problem Understanding the expressive power and limitations of normalizing flows.
method Theoretical framework for well-conditioned coupling-based normalizing flows and volume-preserving flows.
result RealNVP is distributionally universal, but volume-preserving flows are not.
Probabilistic optimal power flow (POPF) is an important analytical tool to ensure the secure and economic operation of power systems. POPF needs to solve enormous nonlinear and nonconvex optimization problems. The huge computational burden has become the major bottleneck for the practical application. This paper presen…
Copula-based normalizing flows improve flexibility and stability for heavy-tailed data.
problem Limited expressive power of vanilla normalizing flows.
method Generalize base distribution to copula for more accurate representation of target distribution.
result Copula-based normalizing flows improve flexibility, stability, and effectiveness for heavy-tailed data.
Unified model explains market dynamics, linking order flow, volatility, and impact.
problem Understanding the dynamics of order flow, market impact, and volatility in financial markets.
method Proposes a microstructural model using Hawkes processes to distinguish core orders and reaction flow, and analyzes their scaling limits.
result Estimates the persistence parameter H0 and finds it consistent with market impact and volatility properties. Study on Bitcoin transaction flows and holding times, revealing multifractal and power-law distributions.
problem Characterizing the temporal behavior and variability of Bitcoin transactions and holding times.
method Analysis of Bitcoin transaction data, including holding-time distributions, multiscaling, and multifractality.
result Found multifractal and power-law distributions in Bitcoin transaction flows and holding times, with significant variations in holding times.
In this paper, we study the convexity, interior gradient estimate, Liouville type theorem and asymptotic behavior at infinity of translating solutions to mean curvature flow as well as the nonlinear flow by powers of the mean curvature.
In this article we analyze totally periodic pseudo-Anosov flows in graph three manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a finite power of the free homotopy class of a closed orbit of the flow. We show …
Study on Monge-Ampère equations with polynomial growth rates.
problem Analyzing solutions to Monge-Ampère equations with polynomial right-hand sides.
method Utilizing polynomial growth analysis to study regularity and growth rates of solutions.
result Translators for sub-affine-critical curvature flows are smooth and convex with specific growth rates.
This paper concerns the evolution of a closed hypersurface of dimension n(≥2) in the Euclidean space Rn+1 under a mixed volume preserving flow. The speed equals a power β(≥1) of homogeneous, either convex or concave, curvature functions of degree one plus a mixed volume preserving term, incl…
We study the motion of an n-dimensional closed spacelike hypersurface in a Lorentzian manifold in the direction of its past directed normal vector, where the speed equals a positive power p of the mean curvature. We prove that for any p∈(0,1], the flow exists for all time when the Ricci tensor of the ambient s…
Study shows decay of correlations on specific types of flows.
problem Analyzing decay of correlations in specific flow types.
method Asymptotic expansion of correlation function on Abelian covers.
result Established an expansion in inverse powers of time.
In this paper, we prove the existence of classical solutions of the Dirichlet problem for a class of quasi-linear elliptic equations on unbounded domains like a cone or a U-type domain. This problem comes from the study of mean curvature flow and its generalization, the flow by powers of mean curvature. Our approach is…
We consider inverse curvature flows in the (n+1)-dimensional Euclidean space, n≥2, expanding by arbitrary negative powers of a 1-homogeneous, monotone curvature function F with some concavity properties. We obtain asymptotical roundness, meaning that circumradius minus inradius of the flow hypersurfaces decay…