Generalizes differentiation under integral sign to submanifolds with corners.
problem Closing a gap in mathematical literature for evolving submanifolds with corners.
method Proves generalizations of the Reynolds Transport Theorem for submanifolds with corners.
result Provides a unified treatment of integral theorems for unbounded cases.
The study characterizes straight-line flows in dynamic measure transport.
problem Tackles the challenge of designing flows that are easy to integrate.
method Characterizes straight-line flows using a PDE and Reynolds tensor.
result Characterizes affine-in-time interpolants and necessary conditions for flow geometry.
The goal of this investigation was to overcome limitations of a persistency analysis, introduced by Benoit Mandelbrot for fractal Brownian processes: nondifferentiability, Brownian nature of process and a linear memory measure. We have extended a sense of a Hurst factor by consideration of a phase diffusion power law. …
Differential chains are a proper subspace of de Rham currents given as an inductive limit of Banach spaces endowed with a geometrically defined strong topology. Boundary is a continuous operator, as are operators that dualize to Hodge star, Lie derivative, pullback and interior product. Partitions of unity exist in thi…
Study uses reinforcement learning to optimize metachronal paddling at low Reynolds number.
problem Optimizing metachronal paddling strategies for efficient swimming at low Reynolds numbers.
method Applied reinforcement learning to a swimmer model with varying paddle spacings.
result The reinforcement learning algorithm selects a back-to-front metachronal wave-like stroke as the most efficient, regardless of the number of paddles.
Data-driven methods for improving turbulence modeling in Reynolds-Averaged Navier-Stokes (RANS) simulations have gained significant interest in the computational fluid dynamics community. Modern machine learning algorithms have opened up a new area of black-box turbulence models allowing for the tuning of RANS simulati…
Proposes a new model for RANS simulations with uncertainty.
problem Uncertainty in Reynolds-averaged Navier-Stokes simulations.
method Data-driven closure model with aleatoric uncertainty, Bayesian formulation, sparse indirect data.
result Accurate probabilistic predictions, even in regions of model error.
Paper applies fluid dynamics to stock market behavior.
problem Understanding stock market dynamics using physical principles.
method Uses Stokes law to model stock market as fluid system.
result Stock market dynamics can be explained by physical properties.
Optimal transport explored on a specific geometric space.
problem Optimal transport problem in sub-Lorentzian Heisenberg group.
method Synthetic metric spacetime structure analysis and sub-Lorentzian version of Brenier's theorem.
result Established sub-Lorentzian version of Brenier's theorem and derived Monge-Ampère equation.
Established a transport theorem for nonconvecting domains on an embedded manifold.
problem Transport theorems for nonconvecting domains evolving on an embedded manifold.
method Used geometric measure theory and the divergence theorem to prove the theorem.
result Proved a transport theorem for nonconvecting domains on an embedded manifold.
The article resolves complex structures in transport twistor spaces, proving a Newlander-Nirenberg theorem.
problem Degenerate complex structures in transport twistor spaces.
method Holomorphic blow-down structure maps to resolve degeneracy and gain insight into complex geometry.
result Global and local β-maps for various metrics, proving a Newlander-Nirenberg theorem for degenerate complex structures.
New theorem shows nearly spherical manifolds can be mapped from spheres.
problem Generalizing Caffarelli's theorem to nearly spherical manifolds.
method Optimal transport map on the sphere, stability result.
result Every nearly spherical manifold can be mapped from a sphere.
Neural network predicts turbulence near-wall regions efficiently.
problem Reducing computational cost in turbulent flow simulations.
method Fully-convolutional neural network trained on DNS data.
result FCN predicts velocity fluctuations at y+=50 with less than 20% error. The paper proves a convergence theorem for Wiener measures on holonomy groups.
problem Understanding convergence of Wiener measures on holonomy groups.
method Using stochastic parallel transports along convergent metric connections.
result Proves a convergence theorem for push-forward Wiener measures on holonomy groups.
Defines parallel 2-transport and 2-group bundles, proving new theorems.
problem Understanding parallel transport in higher dimensions.
method Introduces a new 2-category of 2-group torsors and defines parallel 2-transport as a 2-functor.
result Proves non-Abelian Stokes and Ambrose-Singer theorems for 2-transport.
The paper reviews advances in estimating and understanding optimal transport maps.
problem Estimating and understanding optimal transport maps from samples.
method Recent advances in statistical inference for optimal transport maps.
result Developed limit theorems for the optimal transport map using samples.
The paper proves a global geometric formula for volume holonomy in gauge theory.
problem Describing higher parallel transport in classical principal bundle theory.
method Global geometric approach to parallel transport on surfaces and volumes.
result Global formula for volume holonomy and gauge invariance.
Researchers found counterexamples to conjectures about optimal transport maps on curved spaces.
problem Extending Caffarelli's contraction theorem to curved spaces.
method Constructing counterexamples to precise conjectures.
result Found counterexamples to Milman's conjectures about optimal transport maps on curved spaces.
FEAT estimates free energy using adaptive transports.
problem Estimating free energy across scientific domains.
method Uses learned transports and stochastic interpolants.
result Provides consistent, minimum-variance estimators.
We give a description of the boundary of a complex of free factors that is analogous to E. Klarreich's description of the boundary of a curve complex. The argument uses the geometry of folding paths developed by Bestvina and Feighn as well as structural results about very small trees developed by Coulbois, Hilion, Lust…
Deep learning improves accuracy of RANS simulations for airfoils.
problem Improving accuracy of Reynolds-Averaged Navier-Stokes simulations for airfoils.
method Used a modernized U-net architecture and evaluated various trained neural networks.
result Achieved a mean relative pressure and velocity error of less than 3% across various airfoil shapes.
Novel proof shows continuity of optimal transport feasible set mapping.
problem Continuity of feasible set mapping in optimal transport problems.
method Presented a novel and shorter proof of continuity.
result Established continuity of the feasible set mapping.
Egorov's theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first order transversally elliptic operator with the (classical) evolution of it…
Study MinMax methods for optimization problems, including optimal transport.
problem Optimization problems, especially optimal transport.
method MinMax framework, regularization, neural networks, approximation theorems.
result Justification of neural networks for solving optimization problems.
The rectified flow method is analyzed for its statistical properties.
problem Theoretical support for rectified flow methods is lacking.
method Empirical analysis of rectified flow's statistical properties using regression and density estimation.
result Convergence rates for rectified flow estimators are faster than for nonparametric regression and density estimation.
The paper is accompanying "A general Duality Theorem for the Monge-Kantorovich Transport Problem". We explain the methods used in this article in an elementary setting and present two examples complementing the results obtained therein.
Optimizes transport on submanifolds for curvature inequalities.
problem Proving Michael-Simon-Sobolev inequalities in manifolds with intermediate Ricci curvature bounds.
method Generalizes optimal transport theory to submanifolds and applies to curvature inequalities.
result Proves a variant of the Michael-Simon-Sobolev inequality in manifolds with nonnegative intermediate Ricci curvatures.
A neural network models pressure-Hessian from local velocity gradients in turbulent flows.
problem Modeling the pressure-Hessian from local velocity gradients in turbulent flows.
method Tensor basis neural network (TBNN) trained on DNS data.
result Neural network accurately captures key alignment statistics of the pressure-Hessian tensor.
New bounds for statistical entropic optimal transport with subgaussian measures.
problem Establishing statistical bounds for entropic optimal transport.
method Proving sample complexity and central limit theorem for entropic OT.
result Improved convergence rate and central limit theorem for empirical measures.
The paper introduces surface signatures for irregular surfaces and rough surfaces.
problem Characterizing and integrating highly irregular paths and surfaces.
method Introducing surface signatures and proving extension theorems.
result Surface signatures are universal for surface holonomy and rough surfaces.
On compact manifolds which are not simply connected, we prove the existence of "fake" solutions to the optimal transportion problem. These maps preserve volume and arise as the exponential of a closed 1 form, hence appear geometrically like optimal transport maps. The set of such solutions forms a manifold with dimensi…
We study very small trees from the point of view of reducing systems of free factors, which are analogues of reducing systems of curves for a surface lamination; a non-trivial, proper free factor $F \leq \FN$ reduces T if and only if F acts on some subtree of T with dense orbits. We characterize those trees, call…
The article approximates solutions to the Beltrami equation using similarity surfaces.
problem Approximating solutions to the Beltrami equation.
method Constructing similarity surfaces from polygons and analyzing their conformal uniformization.
result Holomorphic dependence of Christoffel symbols on polygons and convergence to a specific affine connection.
Solves open problem on simple surfaces with novel twistor correspondence.
problem Existence of nontrivial holomorphic vector bundles on simple surfaces.
method Novel twistor correspondence, Nash-Moser inverse function theorem, and microlocal analysis.
result Simple surface twistor space supports no nontrivial holomorphic vector bundles.
Optimal transport on SPD matrices improves domain adaptation for BCI.
problem Improving domain adaptation between two domains using SPD matrices.
method Modelled domain difference as diffeomorphism, used polar factorization theorem for optimal transport, applied weighted Riemannian mean.
result Demonstrated state-of-the-art performance on BCI data sets.
Machine learning predicts wind pressures around circular cylinders efficiently.
problem Predicting wind pressures around circular cylinders using traditional methods is costly and time-consuming.
method Trained GBRT models using Reynolds number, turbulence intensity, and circumferential angle as inputs.
result GBRT models accurately predict wind pressures for a wide range of Reynolds and turbulence intensities.
We show that stochastic interpolation flow maps are Lipschitz with a sharp constant.
problem High dimensional sampling and transport problems.
method Investigating stochastic interpolation flow for generating data samples.
result Stochastic interpolation flow maps are Lipschitz with a sharp constant matching optimal transport maps.
The paper examines soliton surfaces using a parallel transport frame field in 4D space.
problem Geometric properties of soliton surfaces associated with the Betchov-Da Rios equation.
method Parallel transport frame field approach in four-dimensional Euclidean space.
result Characterization of soliton surfaces as flat, minimal, semi-umbilic, or Wintgen ideal.
Paper predicts turbulent flows using physics-informed deep learning.
problem Predicting turbulent flows from fluid simulations.
method Hybrid approach combining RANS and LES with trainable spectral filters and U-net.
result Significant reduction in prediction error for 60 frames ahead.
Survey revisits Bachelier and Dupire, highlighting optimal transport's role.
problem Finding arbitrage-free models calibrated to volatility surfaces.
method Revisits mathematical finance principles, uses optimal transport results.
result Optimal transport provides rigorous foundations for Dupire's model.
We study optimal transportation with the quadratic cost function in geodesic metric spaces satisfying suitable non-branching assumptions. We introduce and study the notions of slope along curves and along geodesics and we apply the latter to prove suitable generalizations of Brenier's theorem of existence of optimal ma…
Study optimal transport on null hypersurfaces and null energy condition.
problem Optimal transport degeneracy on null hypersurfaces.
method Developed tools to characterize null energy condition using convexity properties of entropy.
result Optimal transport characterization of null energy condition.
Quantitative metric spaces study function shapes and sphere diameters.
problem Understanding function shapes and sphere diameters in metric spaces.
method Quantitative analysis of transport-rays decompositions using localization method.
result Bounding the deficit between manifold and sphere diameters.
New findings on optimal transport gradient for generative models, addressing numerical instabilities.
problem Numerical instabilities in training Wasserstein Generative Adversarial Networks (WGAN).
method Valid differentiation theorem for entropic regularized transport, semi-discrete gradient formulation, and optimization algorithm.
result Existence of optimal transport gradient for generative models under specified conditions.
The paper optimizes estimating transport maps between distributions.
problem Estimating optimal transport maps between distributions.
method Plugin approach using optimal couplings and extensions.
result Minimax optimality of the proposed estimators.
New theorem connects probabilistic permanental point processes to Monge-Ampère equation.
problem Probabilistic interpretation of Monge-Ampère equation boundary value problem.
method Large deviation principles and optimal transport theory.
result Explicit rate function for permanental point processes large deviation.
Establishes a Lorentzian Lasry-Lions regularization theorem for functions on globally hyperbolic spacetimes.
problem Optimal transport with C1,1 regularizing pairs method Local semiconcavity and future-directed timelike superdifferentials
result Derives C1,1 regularizing pairs for optimal transport under general assumptions Novel approach to OT using kernel mean embeddings controls overfitting and achieves dimension-free sample complexity.
problem Consistently estimate optimal transport plan from samples.
method Pose OT as learning kernel mean embedding, employ MMD regularization.
result ε-optimal recovery of transport plan and map with dimension-free sample complexity.