New method uses entropy dissipation to prove isoperimetric inequalities.
problem Proving isoperimetric inequalities in geometric settings.
method Information-theoretic approach based on entropy dissipation under heat flow.
result New proof of Euclidean isoperimetric inequality with sharp constant.
A new framework describes dissipation using a metriplectic 4-bracket.
problem Describing dissipation in a way that preserves energy and entropy.
method Using a metriplectic 4-bracket, a quantity like the Poisson bracket with symmetries motivated by Riemannian curvature.
result The metriplectic 4-bracket dynamics includes all known previous binary bracket theories for dissipation.
Paper introduces a new measure combining entropy and Gini index.
problem Quantifying complexity in socio- and econo-physics.
method Generalizes entropy using Gini index and Lorenz curve transformation.
result Supports quantifying complexity in socio- and econo-physics.
By means of a space-time Wasserstein control, we show the monotonicity of the W-entropy functional in time along heat flows on possibly singular metric measure spaces with non-negative Ricci curvature and a finite upper bound of dimension in an appropriate sense. The associated rigidity result on the rate of dissipatio…
Study on convergence of SDEs using entropy methods.
problem Analyzing convergence of stochastic differential equations.
method Applied Lyapunov method to Fokker-Planck equation with weighted relative Fisher information.
result Exponential convergence of probability density function to invariant distribution in L1 distance. Develops neural networks that follow thermodynamics principles.
problem Learning physical systems from data while respecting thermodynamics.
method Uses feedforward neural networks and the GENERIC formalism to enforce metriplectic structure.
result Predictions comply with first and second principles of thermodynamics.
Dissipative SymODEN learns dynamics with dissipation and control from data.
problem Learning dynamics with dissipation and control from observed data.
method Dissipative SymODEN encodes port-Hamiltonian dynamics into a deep learning architecture.
result The learned model reveals key aspects of the system, such as inertia, dissipation, and potential energy.
HCLM framework uses entropy regularization for open learning systems.
problem Real-world AI challenges and limitations of deep learning.
method Dynamical and information-theoretic framework with entropy regularization.
result Geometric entropy surrogates, especially log-determinant covariance entropy, induce stronger and more stable information forces.
Paper adds Fisher Information to mean field optimization for faster convergence.
problem Mean field optimization in neural networks training.
method Developed energy-dissipation method and gradient flow on probability space.
result Marginal distributions converge exponentially to minimizer.
Unified framework for analyzing gradient flows of measures with exponential decay of entropy.
problem Analyzing exponential decay of entropy functionals in gradient flows of measures.
method Characterization of global exponential decay behaviors using Hellinger-Kantorovich geometry, shape-mass decomposition, and Polyak-Łojasiewicz-type inequalities.
result Unified theoretical framework for gradient flows with complete analysis of exponential decay behaviors.
The paper develops a new probabilistic framework for denoising diffusion models using free entropy and stochastic analysis.
problem Developing a mathematical framework for denoising diffusion models in noncommutative settings.
method Formulating diffusion and reverse processes governed by operator-valued stochastic dynamics, using tools from free stochastic analysis.
result Establishing an information-geometric link between entropy production, transport, and deconvolution.
The paper integrates dissipative and curl forces using geometric methods.
problem Incorporating dissipative forces into curl forces for non-conservative systems.
method Geometric metriplectic approach, Herglotz principle, generalized Euler-Lagrange equation, Galley's method.
result Natural formulations for Lagrangian and Hamiltonian dynamics of non-conservative systems.
Geometrically describes Jacobi equations for field theories with dissipation.
problem Describing field theories with dissipation geometrically.
method Prolongation of the Lagrangian on a k-cosymplectic formulation to describe Jacobi equations and a modified Lagrangian for variational formulation.
result Variational formulation of field theories with dissipation.
Develops geometric framework for dissipative field equations.
problem Dissipative field equations and their geometric analysis.
method Canonical k-contact manifolds, k-contactifications, splitting results, regularity conditions, criteria for PDEs. result Explicit Hamiltonian descriptions for various nonlinear PDEs.
We study velocity correlations induced by diffusion and dissipation in a simple dissipative dynamical system. We observe that diffusion, as a result of time reversible microscopic processes, leads to correlations with different spatial parity from those caused by dissipation, consisting of time irreversible microscopic…
New blurring diffusion models bridge heat dissipation and denoising.
problem Developing a new generative modeling approach.
method Connecting blurring to Gaussian diffusion with non-isotropic noise.
result Proposed Blurring Diffusion Models offer the best of both Gaussian denoising and inverse heat dissipation.
We study the dynamical behaviors of degenerate stochastic differential equations (SDEs). We select an auxiliary Fisher information functional as the Lyapunov functional. Using generalized Fisher information, we conduct the Lyapunov exponential convergence analysis of degenerate SDEs. We derive the convergence rate cond…
Unified geometric framework for integrability of conservative and dissipative systems.
problem Unified definition of integrability for both conservative and dissipative systems.
method Introducing Jacobi-Haantjes manifolds and contact-Haantjes manifolds to unify definitions.
result Equivalence of integrability in contact Hamiltonian systems and existence of Abelian extended Haantjes algebra.
Geometric framework for dissipative systems on Lie algebroids.
problem Formulating geometric framework for dissipative systems.
method Herglotz-type variational principle on Lie algebroids.
result Recover classical equations as special cases.
Derives stochastic and dissipative dynamics preserving Gibbs measure.
problem Understanding and deriving structure-preserving stochastic systems.
method Extension of Hamilton-Pontryagin principle, symmetry reduction, and inclusion of dissipation.
result New derivation of double-bracket dissipation.
Researchers present and compare different representations of dissipative Hamiltonian DAE systems.
problem Understanding and transforming dissipative Hamiltonian DAE systems.
method Global geometric and algebraic points of view, translations between representations, characterizations, and numerical methods for computing structural information.
result A general DAE system can be transformed into a dissipative Hamiltonian or port-Hamiltonian DAE system.
Geometric integrator preserves coadjoint orbits in dissipative systems.
problem Preserving coadjoint orbits in dissipative mechanical systems.
method Adapted discrete variational integrators for forced Euler-Poincaré and Lie-Poisson systems.
result Preserves coadjoint orbits exactly, improving over general-purpose methods.
The paper applies thermodynamics to financial markets to prove no-arbitrage constraints.
problem No arbitrage in financial markets under price impact.
method Stochastic thermodynamics applied to financial trading cycles.
result Proves any round-trip trading strategy yields non-positive expected profit.
Improved efficiency in HMC samplers reduces dissipative behavior.
problem Reducing dissipative behavior in HMC samplers.
method Variable integration time and partial velocity refreshment.
result Efficiency improved by a √κ factor in Wasserstein-2 distance.
In this paper we introduce and study a new kind of hyperbolic geometric flows --dissipative hyperbolic geometric flow. This kind of flow is defined by a system of quasilinear wave equations with dissipative terms. Some interesting exact solutions are given, in particular, a new concept-- hyperbolic Ricci soliton is int…
We develop a new geometric framework suitable for dealing with Hamiltonian field theories with dissipation. To this end we define the notions of k-contact structure and k-contact Hamiltonian system. This is a generalization of both the contact Hamiltonian systems in mechanics and the k-symplectic Hamiltonian syst…
On a Riemannian manifold (M,g) we consider the k+1 functions F1,...,Fk,G and construct the vector fields that conserve F1,...,Fk and dissipate G with a prescribed rate. We study the geometry of these vector fields and prove that they are of gradient type on regular leaves corresponding to F1,...,Fk. B…
Derives sub-Riemannian Ricci curvature for various manifolds.
problem Calculating Ricci curvature in sub-Riemannian geometry.
method Generalized Gamma z calculus and z--Bochner's formula. result Analytical bounds for sub-Riemannian curvature dimension and log-Sobolev inequalities.
In this paper we formulate a geometric theory of the mechanics of growing solids. Bulk growth is modeled by a material manifold with an evolving metric. Time dependence of metric represents the evolution of the stress-free (natural) configuration of the body in response to changes in mass density and "shape". We show t…
Develops a mean-field theory for multi-head self-attention under cross-entropy training.
problem Mean-field analysis of multi-head self-attention under cross-entropy training.
method Mean-field theory for a simplified single-layer causal multi-head self-attention model.
result Proves a static finite-head approximation bound for the optimal risk.
FNOs learn solution operators of dissipative equations efficiently via spectral methods.
problem Learning and approximation of solution operators for dissipative equations.
method Introducing spectral methods and deriving FNO approximation bounds and sample complexity guarantees.
result Polynomial sample complexity guarantees for FNOs learning solution operators of dissipative equations.
In two recent papers necessary and sufficient conditions for a given system of second-order ordinary differential equations to be of Lagrangian form with additional dissipative forces were derived. We point out that these conditions are not independent and prove a stronger result accordingly.
Develop contact Tulczyjew formalism for dissipative dynamics on skew algebroids.
problem Dissipative dynamics on skew algebroids
method Contact Tulczyjew formalism
result Intrinsic explanation of contact term and Euler-Lagrange-Herglotz equations
A new method called MCLMC avoids dissipation in sampling from canonical distributions.
problem Sampling from canonical distributions without dissipation.
method Microcanonical Langevin Monte Carlo (MCLMC) as a dissipation-free system of SDE.
result MCLMC converges faster than HMC for lattice φ^4 models.
DiMS sampler explores neural network loss minima via dissipative dynamics.
problem Sampling reparameterization invariant solutions in neural networks.
method Dynamical system based on kinetic energy with dissipative friction.
result DiMS sampler samples exactly from minimum level sets.
New method preserves convergence rates in gradient-based optimization.
problem How to discretize gradient-based optimization systems while preserving stability and convergence rates.
method Geometric framework for dissipative symplectic integration.
result Dissipative symplectic integrators preserve rates of convergence up to a controlled error.
D-LinOSS models learn to dissipate energy, improving performance on long-range tasks.
problem Representational limitations of LinOSS models in long-range reasoning.
method Introducing Damped Linear Oscillatory State-Space models (D-LinOSS) that learn to dissipate latent state energy on arbitrary time scales.
result D-LinOSS consistently outperforms previous LinOSS methods on long-range learning tasks, achieving faster convergence and reducing hyperparameter search space.
Quantum systems learn like machine learning models, influenced by dissipation.
problem Understanding how quantum systems learn and evolve.
method Hydrodynamical formulation of quantum mechanics, gradient descent model, empirical demonstration.
result Quantum systems follow a disrupted gradient descent model, influenced by dissipation.
EuSN uses Euler discretization for stable, non-dissipative reservoir computing.
problem Designing stable and efficient reservoir computing models.
method Forward Euler discretization and antisymmetric recurrent matrices.
result EuSN outperforms standard RC models in long-term memory tasks and time-series classification.
By treating the financial market as a thermodynamic system, we establish a one-to-one correspondence between thermodynamic variables and economic quantities. Measured by the expected loss under the worst-case scenario, financial risk caused by model uncertainty is regarded as a result of the interaction between financi…
In this paper, we extend the well-known Noether theorem for Lagrangian systems to contact Lagrangian systems. We introduce a classification of infinitesimal symmetries and obtain the corresponding dissipated quantities. We notice that in contact dynamics, the existence of infinitesimal symmetries does not produce conse…
Introduces a new bracket for multicontact geometry and applies it to field theories.
problem Developing a new mathematical structure for multicontact geometry.
method Introducing a graded Jacobi bracket and multisymplectization.
result Established a new bracket that extends contact geometry concepts.
Develops Hamilton-Jacobi theory for non-conservative field theories in k-contact geometry.
problem Analyzes non-conservative field theories, especially dissipative systems.
method Introduces evolution k-contact k-vector fields and develops two Hamilton-Jacobi theories.
result Recover ordinary contact Hamilton-Jacobi theory as k=1, and enlarges application range.
DeepONet learns operators for PDEs with varying parameters and initial conditions.
problem Learning operators for partial differential equations with different parameters or initial conditions.
method DeepONet uses a Branch net and Trunk net to minimize error between evaluated and expected outputs, incorporating a scalar auxiliary variable approach for energy dissipation.
result DeepONet can accurately approximate operators for PDEs with varying parameters or initial conditions.
LipKernel adds robustness to CNNs by enforcing Lipschitz bounds.
problem Improving robustness of CNNs in real-time applications.
method Dissipative layers parameterized by LMIs and 2-D Roesser model.
result Orders of magnitude faster run-time compared to state-of-the-art methods.
Study non-asymptotic Langevin Monte Carlo for Gibbs distributions.
problem Sampling from Gibbs distributions with dissipative potentials.
method Langevin-type algorithms based on Liptser--Shiryaev theory and Poincaré inequalities.
result Upper bound on 2-Wasserstein distance for accurate approximation.
GOPO optimizes large models in Hilbert space, avoiding Kullback-Leibler's curvature.
problem Optimizing large language models with Kullback-Leibler divergence's curvature issues.
method GOPO uses Hilbert space L2(pi_k) with orthogonality constraints and a work-dissipation functional.
result GOPO achieves competitive generalization with stable gradient dynamics and entropy preservation.
Paper bridges quantum and classical mechanics for open systems.
problem Quantum open systems with bi-Lindblad structure.
method Develops a bridge between bi-Hamiltonian structures and GKSL formalism, introducing contact-compatible Lindblad generators.
result Provides a mathematical mechanism for semiclassical limit of quantum open systems.