A new model estimates complex densities without explicit normalizing constants.
problem Accurately estimating the normalizing constant of high-dimensional energy functions.
method Autoregressive Energy Machine (AEM) learns an unnormalized density and an importance-sampling estimate of the normalizing constant.
result Achieves state-of-the-art performance on density-estimation tasks.
Quantum vacuum energy (Casimir energy) is reviewed for a mathematical audience as a topic in spectral theory. Then some one-dimensional systems are solved exactly, in terms of closed classical paths and periodic orbits. The relations among local spectral densities, energy densities, global eigenvalue densities, and tot…
The paper examines geometric curvatures in generalized Riemannian spaces.
problem Understanding the physical meaning of scalar curvatures in generalized Riemannian spaces.
method Developed Madsen's formulae for pressures and energy-densities, analyzed with different concepts of generalized Riemannian spaces.
result Linearities of energy-momentum tensor, pressure, energy-density, and state-parameter are examined.
Efficiently samples and learns densities with symmetries using equivariant methods.
problem Efficiently sampling and learning densities with symmetries.
method Equivariant Stein Variational Gradient Descent (SVGD) and equivariant energy based models.
result Improves and scales up training of energy based models.
The paper classifies biharmonic quadratic maps between spheres, proving their energy density properties.
problem Classifying non-harmonic biharmonic quadratic forms between spheres.
method Proving non-harmonic biharmonic quadratic forms have constant energy density and classifying them.
result Non-harmonic biharmonic quadratic forms have constant energy density (m+1)/2. The paper introduces a new energy density function and proves Liouville type theorems for various maps.
problem Proving Liouville type theorems for holomorphic, harmonic, and pluri-harmonic maps.
method Introducing a new energy density function and deriving Hessian estimates.
result No non-constant holomorphic map exists between certain Hermitian manifolds with specific curvature conditions.
Kernel ridge regression is used to approximate the kinetic energy of non-interacting fermions in a one-dimensional box as a functional of their density. The properties of different kernels and methods of cross-validation are explored, and highly accurate energies are achieved. Accurate {\em constrained optimal densitie…
We show that the energy density of critical points of a class of conformally invariant variational problems with small energy on the unit 2-disk B_1 lies in the local Hardy space h^1(B_1). As a corollary we obtain a new proof of the energy convexity and uniqueness result for weakly harmonic maps with small energy on B_…
Study on JNR monopoles, focusing on their spectral curves and energy density.
problem Understanding JNR monopoles and their spectral curves.
method Analysis of spectral curves, rational maps, and holomorphic spheres.
result Established conditions for a spectral curve to be a JNR monopole and derived a formula for energy density at infinity.
Stable solutions to a specific equation are one-dimensional.
problem Stability and dimensionality of solutions to the Allen-Cahn equation.
method Analysis of stable solutions with bounded energy density.
result Stable solutions to the Allen-Cahn equation are one-dimensional.
New method trains deep neural networks for non-interacting kinetic-energy functionals in DFT.
problem Lack of exact relationship between electron density and non-interacting kinetic energy.
method Variational principle to regularize machine-learned density functionals.
result Excellent results on kinetic-energy functionals for various systems.
A new training method for normalizing flows without samples.
problem Training normalizing flows without samples but with energy functions.
method Interpolates energy functions to find a transport vector field.
result Optimizes transport vector field and energy function to satisfy continuity equation.
We prove that capillary surfaces converge to a specific energy density as the angle approaches zero.
problem Understanding the limiting behavior of capillary surfaces as the angle approaches zero.
method Rigorous limiting analysis and application to curvature estimates.
result Capillary area-density converges to the Weiss energy density as the angle tends to zero.
iEFM trains CNF models from unnormalized densities efficiently.
problem Training generators from energy functions or unnormalized densities.
method Iterated energy-based flow matching (iEFM) with simulation-free objective.
result iEFM outperforms existing methods in probabilistic modeling.
Proves density and mass theorems for specific initial data sets.
problem Initial data sets with boundary in spacetime.
method Harmonic asymptotics and dominant energy condition.
result Spacetime positive mass theorem for initial data sets with apparent horizon boundary.
The study shows how energy density of harmonic maps dominates in n-Fuchsian fibers, leading to unique minimal surfaces.
problem Understanding energy density and topological invariants in n-Fuchsian fibers of Higgs bundles. method Establishing an algebraic inequality generalizing a GIT theorem to prove energy density domination.
result Energy density of harmonic maps dominates in n-Fuchsian fibers, leading to unique minimal surfaces. DEEN learns energy and score functions from complex data.
problem Challenges in density estimation for high-dimensional data.
method Inference-free hierarchical framework using score matching and multilayer perceptrons.
result DEEN successfully learns energy and score functions from synthetic and high-dimensional data.
We give a derivation of the Einstein equation for gravity which employs a definition of the local energy density of the gravitational field as a symmetric second rank tensor whose value for each observer gives the trace of the spatial part of the energy-stress tensor as seen by that observer. We give a physical motivat…
Study on the Lp behavior of helix curves' energy density.
problem Analyzing the Lp asymptotics of helix curves' Möbius energy density. method Contour integration and Laurent expansion near poles.
result Established the precise Lp blowup rate of the Möbius energy density. RNE provides a flexible framework for diffusion models, enabling inference-time control and energy-based training.
problem Insufficient knowledge of marginal densities in diffusion models.
method Introduces Radon-Nikodym Estimator (RNE) to reveal the connection between marginal densities and transition kernels.
result RNE delivers strong results in inference-time control and energy-based diffusion training.
iDEM generates samples from Boltzmann densities without data.
problem Generating statistically independent samples from unnormalized distributions.
method Iterative algorithm using energy and gradient for diffusion-based sampler training.
result iDEM achieves state-of-the-art performance and trains faster than existing methods.
Adaptive multi-stage density ratio estimation improves learning of latent space EBM.
problem Learning energy-based models in latent space is computationally expensive and challenging.
method Adaptive multi-stage density ratio estimation using NCE to bridge the gap between prior and posterior densities.
result The method enables more expressive prior models and sharpens the latent space EBM.
For Ginzburg-Landau vortices, energy quantization holds only when density is less than 2.
problem Energy quantization in Ginzburg-Landau vortices for higher dimensions.
method Analyzing normalized energy measures and vorticity sets.
result Energy quantization only holds when density is less than 2.
When working with asymptotically hyperbolic initial data sets for general relativity it is convenient to assume certain simplifying properties. We prove that the subset of initial data sets with such properties is dense in the set of physically reasonable asymptotically hyperbolic initial data sets. More specifically, …
The paper tackles semi-supervised learning on point clouds using PDE methods.
problem Extend labels to an entire data set on point clouds.
method Minimizing constrained discrete p-Dirichlet energy, connecting to continuum p-Dirichlet energy, applying PDE methods like pseudo-spectral methods. result Consistency of the numerical scheme in the large data limit for density estimation methods.
Bidirectional bounds stabilize training of energy-based models.
problem Training energy-based models is difficult and prone to instability.
method Propose bidirectional bounds linking to gradient penalty and Jacobi-determinant estimator.
result Significant stabilization and high-quality density estimation achieved.
Researchers created an accurate kinetic energy functional for materials modeling.
problem Lack of accurate analytic kinetic energy functionals for large-scale ab initio materials modeling.
method Interpretative machine learning of crystal cell-averaged kinetic energy densities guided by a hybrid Gaussian process regression - neural network (GPR-NN) method.
result Constructed an analytic kinetic energy functional that reproduces Kohn-Sham DFT energy-volume curves with sufficient accuracy.
A new Möbius invariant discretization and decomposition of the Möbius energy is proposed.
problem Lack of Möbius invariant discretization and decomposition in existing discrete Möbius energy.
method Proposed a new discretization of Möbius energy that is Möbius invariant and can be decomposed into Möbius invariant components.
result The proposed discretization and decomposition maintain Möbius invariance and converge to the original components in the continuum limit.
Derives continuum model from discrete ε-graphs with connectivity functional.
problem Modeling diffusion in networks with varying connectivity.
method Energy-based continuum limit derivation, neural-network reconstruction of connectivity.
result Error between discrete and continuum energies is O(ε), valid even with fluctuations. Differential conservation laws in Lagrangian field theory are usually related to symmetries of a Lagrangian density and are obtained if the Lie derivative of a Lagrangian density by a certain class of vector fields on a fiber bundle vanishes. However, only two field models meet this property in fact. In gauge theory of…
A new method predicts electron density accurately from atom-centered models.
problem Predicting electron density accurately from atom-centered models.
method Gradient-based approach to minimize loss function in an optimized sparse feature space.
result Extremely accurate predictions of electron density and total energies.
Improves GANs by sampling from an energy-based model induced by discriminator scores.
problem Improving the quality of images generated by GANs.
method DDLS (Discriminator Driven Latent Sampling) using the sum of latent prior log-density and discriminator output score.
result Significantly improves Inception Score on CIFAR-10 dataset.
Unified framework for OOD detection using class ratio estimation.
problem Density-based OOD detection is unreliable for OOD images.
method Unified framework that builds energy-based models and employs differing base distributions, directly estimating the density ratio through class ratio estimation.
result Competitive results on OOD image problems compared to recent work.
Machine learning bypasses Kohn-Sham equations for faster DFT calculations.
problem Solving the Kohn-Sham equations for electronic structure problems.
method Directly learning density-potential and energy-density maps for test systems and molecules.
result Improved accuracy and lower computational cost demonstrated for molecular geometries.
The paper proposes a new method for probabilistic load forecasting using Bernstein-Polynomial Normalizing Flows.
problem High variability in short-term load forecasting at the low-voltage level due to fluctuating demand and increasing electrification.
method Flexible conditional density forecasting based on Bernstein polynomial normalizing flows with neural network control.
result Density predictions outperform traditional methods for 24h-ahead load forecasting.
The paper proves that certain maps from complex space to Kahler manifolds are holomorphic.
problem Characterizing harmonic maps from complex space to Kahler manifolds.
method Proves that harmonic maps from Cn to any Kahler manifold must be holomorphic under a specific condition. result Harmonic maps from Cn to Kahler manifolds are holomorphic under an energy density assumption. Persistently trained EBMs generate images and estimate complex densities.
problem Challenges in ML learning for energy-based models, especially non-convergence of MCMC.
method Introduce diffusion data, learn a joint EBM through persistent training with enhanced sampling.
result First simultaneous achievement of stability, post-training image generation, and superior out-of-distribution detection for image data.
Neural network predicts daily power consumption with high accuracy.
problem Middle-term power consumption prediction in the energy sector.
method Incorporates trend, seasonality, and weather conditions in a shallow Neural Network.
result Excellent density forecast results on one-year test set.
Wavelet scattering predicts molecular energies efficiently.
problem Estimating quantum chemical energies of organic molecules efficiently.
method Multiscale invariant dictionaries with wavelet scattering.
result Regression error is comparable to DFT codes but faster.
New method trains EBMs using NFs for more accurate likelihood estimation.
problem Lack of statistical accuracy in EBMs likelihood estimation.
method Uses normalizing flows (NF) to fit an NF to an EBM during training.
result Accurate gradient for EBMs at all times, leading to a fast sampler.
Machine learning is used to approximate density functionals. For the model problem of the kinetic energy of non-interacting fermions in 1d, mean absolute errors below 1 kcal/mol on test densities similar to the training set are reached with fewer than 100 training densities. A predictor identifies if a test density is …
Hitchin representations yield harmonic maps with energy density ≥1.
problem Characterize harmonic maps associated with Hitchin representations.
method Study ρ-equivariant harmonic maps f from a hyperbolic surface to symmetric spaces. result Energy density of f is ≥1, with equality at one point only for specific representations. The study classifies biharmonic submanifolds in a sphere using specific eigenmaps.
problem Characterizing biharmonic submanifolds in a sphere.
method Classification based on bi-eigenmaps and buckling eigenmaps.
result Generalizations of Takahashi's characterization of minimal submanifolds in a sphere.
Uniqueness of stable, non-smooth hypersurfaces with constant anisotropic mean curvature.
problem Identifying stable, non-smooth hypersurfaces with constant anisotropic mean curvature.
method Study of piecewise-smooth hypersurfaces with anisotropic energy, proving uniqueness of the Wulff shape under certain conditions.
result Closed stable equilibrium hypersurfaces are unique and the Wulff shape when the anisotropic energy density is twice continuously differentiable and convex.
Extends Penrose's method to null shells with pressure and energy flux.
problem Constructing null thin shells with arbitrary gravitational/matter content.
method Derive locally Lipschitz metric and coordinate transformation.
result Example of null shell with non-trivial energy density, flux, and pressure in Minkowski space.
CEBMs learn flexible latent mappings from data.
problem Learning flexible latent mappings from data.
method CEBMs decompose joint density into tractable posterior over latent variables.
result CEBMs achieve competitive results in image modeling and latent space predictive power.
Smooth long-time existence proved for Harmonic Ricci Flow on surfaces.
problem Smooth long-time existence of Harmonic Ricci Flow on surfaces with positive genus.
method Proved finite singular time blow-up conditions for energy density and curvature.
result Smooth long-time existence for large coupling constant Harmonic Ricci Flow.
FNFs model parameter-dependent densities by combining a fixed flow with a polynomial parameter-dependent transformation.
problem Learning a separate flow for every parameter configuration is intractable.
method Factorizable Normalizing Flows (FNFs) represent the parameter-dependent density as a fixed flow for a reference configuration and a learnable polynomial transformation factorized over parameters.
result FNFs enable the recovery of the combined effect of multiple parameters without sampling their joint space, providing a scalable and interpretable solution.