The paper analyzes numerical instability in variational flows and proposes a diagnostic method.
problem Numerical instability in variational flows affects sampling, density evaluation, and ELBO estimation.
method Treated variational flows as dynamical systems, used shadowing theory for theoretical guarantees, and developed a diagnostic procedure.
result Despite numerical instability, results from variational flows can be accurate enough for practical applications.
i-flow uses normalizing flows for high-dimensional integration and sampling.
problem High-dimensional integration in science and statistics.
method Normalizing flows for bijective mappings between distributions.
result i-flow outperforms other algorithms for high-dimensional correlated integrals.
Investigates stability of piecewise flat Ricci flow using analysis and simulations.
problem Stability of piecewise flat Ricci flow.
method Linear stability analysis and numerical simulations.
result Adaptations avoided numerical instability and led to convergence to smooth solutions.
Study shows different behaviors of noncompact hypersurfaces under mean curvature flow.
problem Analyzing stability of noncompact hypersurfaces with curvature blowup.
method Numerical overlap method to construct global solutions.
result Existence of near and far classes of initial data leading to distinct behaviors.
Kähler-Ricci flow singularity type is independent of initial metric.
problem Independence of singularity type for Kähler-Ricci flows.
method Analyzing solutions to the Kähler-Ricci flow on numerically effective manifolds.
result The singularity type of solutions is independent of the initial metric.
Numerical method simulates Ricci flow on various 3-manifolds.
problem Simulate Ricci flow on compact 3-manifolds without boundary.
method Piecewise flat numerical method for evolving Ricci flow.
result Converges to known smooth solutions for homogeneous manifolds.
Combines normalizing flows and quasi-Monte Carlo for improved numerical integration.
problem Improving the efficiency of numerical integration methods.
method Uses normalizing flows to approximate distributions and quasi-Monte Carlo for sampling.
result Demonstrates an estimator with significantly lower variance.
A new gradient flow for MMD with closed-form implementation.
problem Existing gradient flows either lack tractable numerical implementation or require strong assumptions.
method Introduces a (de)-regularized Maximum Mean Discrepancy (DrMMD) and its gradient flow.
result Guarantees near-global convergence for a broad class of targets in both continuous and discrete time.
Study proposes curvature flow model for Drosophila dorsal closure.
problem Modeling and understanding Drosophila dorsal closure during embryonic development.
method Curvature-based mathematical model, analysis of maximum-principle and integral-estimates, numerical approximation scheme.
result Established global existence and convergence for the model.
We present numerical visualizations of Ricci Flow of surfaces and 3-dimensional manifolds of revolution. Ricci_rot is an educational tool which visualizes surfaces of revolution moving under Ricci flow. That these surfaces tend to remain embedded in R3 is what makes direct visualization possible. The numerical lessons …
Study Chen's flow of curves in two settings: closed circles and lines, identifying geometric conditions for global behavior.
problem Understanding the global behavior of Chen's flow of curves in two settings.
method Investigated two settings: closed immersed ω-circles and immersed lines with a cocompactness condition. Analyzed geometric conditions and curvature effects.
result Identified conditions ensuring the flow shrinks every initial curve to a point, including a rescaling method.
Variational approximations for curve flows on Riemannian manifolds.
problem Approximating solutions to curvature and elastic flow problems on Riemannian manifolds.
method Variational formulations, finite element approximations, piecewise linear elements, stability analysis.
result Derived schemes can compute rotationally symmetric self-shrinkers and geodesics.
Gradient flow preserves speed for integral Menger curvature curves.
problem Optimizing curves with integral Menger curvature constraints.
method Projected Sobolev gradient flow in Hilbert space.
result Long-time existence and C1,1-bounds for the flow. Researchers develop a numerical method to compute the index of self-shrinkers, finding it to be 5 for the Angenent torus.
problem Computing the index of unstable self-shrinkers in mean curvature flow.
method Numerical method for computing the Morse index of rotationally symmetric self-shrinkers.
result The index of the Angenent torus is 5, with two additional variations found.
This work considers the question of whether mean-curvature flow can be modified to avoid the formation of singularities. We analyze the finite-elements discretization and demonstrate why the original flow can result in numerical instability due to division by zero. We propose a variation on the flow that removes the nu…
We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a di…
We study the evolution of wormhole geometries under Ricci flow using numerical methods. Depending on values of initial data parameters, wormhole throats either pinch off or evolve to a monotonically growing state. The transition between these two behaviors exhibits a from of critical phenomena reminiscent of that obser…
Modeling bone microarchitecture adaptation using geometric flows.
problem Bone microarchitecture adaptation modeling.
method Advection and mean curvature flow model with a sphere as a test case.
result Closed-form solution for sphere under advection and mean curvature flow.
We consider the general Kähler-Ricci flows which exist for all time. The zeroth order control on the flow metric potential for various infinite time singularities is the focus. The possible semi-amplness for numerically effective classes serves as the main motivation.
Generative models tackle incompressible fluid flows by enforcing divergence-free constraints.
problem Simulating incompressible fluid flows with generative models.
method Score-based diffusion models with divergence-free constraint.
result Models can reproduce Kolmogorov turbulence characteristics.
New Ricci flow method for directed graphs with balancing factor.
problem Analyzing asymmetry in directed networks.
method Rigorous formulation of Ricci flow on directed weighted graphs with balancing factor.
result Existence and uniqueness of discrete Ricci flow solutions.
Paper solves trade-off between internalisation and externalisation in stochastic trade flows.
problem Managing risk in stochastic trade flows between internalisation and externalisation.
method Derives almost-closed-form solutions using Almgren-Chriss framework for quadratic execution costs. Uses numerical methods for more general cases. Proposes reinforcement learning as an alternative.
result Almost-closed-form solutions and numerical methods for optimal strategies.
Visualizes deep network feature contributions in images.
problem Understanding information flow in deep networks.
method Forward-Backward approach for feature visualization.
result Numerical results show benefits over existing methods.
We use numerical techniques to study the formation of singularities in Ricci flow. Comparing the Ricci flows corresponding to a one parameter family of initial geometries on S^3 with varying amounts of S^2 neck pinching, we find critical behavior at the threshold of singularity formation.
The paper accelerates gradient flows on probability distributions using optimal control theory.
problem Optimizing probability distributions efficiently.
method Variational formulation and Hamilton's equations for accelerated gradient flows.
result The method achieves accelerated density transport from any initial distribution to a target distribution.
Using the conformally invariant Cotton tensor, we define a geometric flow, the "Cotton flow", which is exclusive to three dimensions. This flow tends to evolve the initial metrics into conformally flat ones, and is somewhat orthogonal to the Yamabe flow, the latter being a flow within a conformal class. We define an en…
Paper proposes efficient training for normalizing flows in Boltzmann generators.
problem Training normalizing flows for Boltzmann generators is computationally challenging and unstable.
method Regression Training of Normalizing Flows (RegFlow) using ℓ2-regression. result RegFlow enables efficient and stable training of normalizing flows for Boltzmann generators.
A new method for optimization in probability space using Newton's flows.
problem Optimization in probability space with information metrics.
method Information Newton's flows, including Fisher-Rao and Wasserstein-2 metrics, with Newton's Langevin dynamics and variational methods.
result Effective numerical implementation and convergence results for the proposed method.
Study on migrating elastic flows of curves across half-planes.
problem Migrating elastic flows of curves from upper to lower half-planes.
method Analytical and numerical construction of migrating elastic flows.
result Construction of various migrating elastic flows.
NSFs learn SDE transition laws for efficient sampling.
problem Efficiently sampling between arbitrary time points in SDEs.
method Conditional normalising flows with architectural constraints.
result Up to two orders of magnitude speed-ups at large time gaps.
A new gradient flow framework for distributionally robust optimization.
problem Optimizing under uncertainty with worst-case distributional constraints.
method Gradient flow theory applied to distributionally robust optimization.
result Practical algorithms for sampling from worst-case distributions.
A robot learns environmental fields using physics-based models and Bayesian methods.
problem Accurately learning complex environmental fields from limited robot measurements.
method Bayesian framework with Gaussian processes to select and update physics-based models in real-time.
result The robot's learned flow field approximates real flow better than prior solutions and data-driven methods.
The paper develops flows for tori and spheres, addressing complex geometries.
problem Learning flows on tori and spheres for complex geometries.
method Recursive flows starting from circles, intervals, or spheres.
result Expressive and numerically stable flows on tori and spheres.
New method uses anisotropic mean curvature flow for contour recognition.
problem Contour recognition in images.
method Coupling anisotropic mean curvature flow with external charges for curve motion.
result Stable numerical approximation for contour recognition.
Study Kähler-Ricci flow on rational homogeneous varieties using algebraic geometry and representation theory.
problem Analyzing the Kähler-Ricci flow on rational homogeneous varieties.
method Combining projective algebraic geometry and representation theory of semisimple Lie groups and Lie algebras.
result Explicit description and computation of solutions and geometric quantities along the flow.
The Willmore flow is well known problem from the differential geometry. It minimizes the Willmore functional defined as integral of the mean-curvature square over given manifold. For the graph formulation, we derive modification of the Willmore flow with anisotropic mean curvature. We define the weak solution and we pr…
Ricci flow simulations show unstable Fubini-Study metrics develop singularities.
problem Understanding the behavior of unstable perturbations in Ricci flow.
method Numerical simulations of Ricci flow starting from unstable Fubini-Study metrics.
result Ricci flow solutions from unstable Fubini-Study metrics develop local singularities.
Theory explains why neural nets better learn Calabi-Yau metrics.
problem Learning Calabi-Yau metrics with neural networks.
method Developed a theory of metric flows in neural network space.
result Finite-width neural networks learn Calabi-Yau metrics better than fixed kernel methods.
sFML learns stochastic dynamical systems from data.
problem Learning unknown stochastic dynamical systems from measurement data.
method sFML extends FML for deterministic systems, using a stochastic flow map composed of deterministic and stochastic sub-maps.
result sFML constructs a stochastic evolution model approximating unknown stochastic systems.
New numerical methods for evolving curves on curved spaces.
problem Evolve curves on Riemannian manifolds efficiently and accurately.
method Variational approximations and numerical schemes for curvature flow, curve diffusion, and elastic flow.
result Effective numerical schemes for geometric evolution equations on Riemannian manifolds.
We map out the moduli space of Lawson symmetric constant mean curvature surfaces in the 3-sphere of genus g>1 by flowing numerically from Delaunay tori with even lobe count via the generalized Whitham flow.
We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially symmetric. For axially symmetric initial surfaces with sufficiently large volume…
We interpret policy optimization as Wasserstein gradient flows and develop efficient algorithms.
problem Unclear mathematical principle of policy optimization in reinforcement learning.
method Interpreting policy optimization as Wasserstein gradient flows, developing efficient algorithms to solve the corresponding discrete gradient flows.
result Policy optimization becomes a convex problem in terms of distribution optimization under specified circumstances.
SeqRF straightens generative model flows to speed up sampling.
problem High global truncation error in ODE-based solvers for generative models.
method SeqRF, a learning technique that straightens the probability flow.
result Significantly improved sampling speed and synthesis quality.
We derive a numerical method for Darcy flow, hence also for Poisson's equation in mixed (first order) form, based on discrete exterior calculus (DEC). Exterior calculus is a generalization of vector calculus to smooth manifolds and DEC is one of its discretizations on simplicial complexes such as triangle and tetrahedr…
Regularizes f-divergences with MMD to analyze Wasserstein flows.
problem Limitations of f-divergences in measures' support. method Rewriting MMD regularization as Moreau envelope in RKHS, analyzing gradients.
result Analysis of Wasserstein flows of MMD-regularized f-divergences. Existence of symmetric critical knots proven for O'Hara's energy family.
problem Existence of symmetric critical knots for O'Hara's knot energy family.
method Proved using Palais' principle of symmetric criticality.
result At least two smooth Eα-critical knots in every torus knot class. The second del Pezzo surface is known by work of Tian-Zhu and Wang-Zhu to admit a unique Kaehler-Ricci soliton. Applying a method described in hep-th/0703057, we use Ricci flow to numerically compute that soliton metric. We numerically compute the value of its Perelman entropy (or Gaussian density).