Generative model improved using Liouville PDE-based sliced-Wasserstein flow.
problem Improving generative models for fair regression.
method Transformed sliced-Wasserstein flow into Liouville PDE-based formalism, handling density estimation with normalizing flows of neural ODE.
result Outperforms in convergence and fairness with reduced variance.
Paper explores Fisher-Rao gradient flows and their kernel approximations.
problem Understanding and analyzing approximations of Fisher-Rao gradient flows.
method Rigorous investigation of Fisher-Rao and Wasserstein type gradient flows, focusing on kernel approximations.
result Proves evolutionary Γ-convergence for kernel-approximated Fisher-Rao flows, providing theoretical guarantees.
This paper provides a mathematical foundation for deep neural networks solving PDEs.
problem Mathematical foundation for deep neural networks solving high-dimensional PDEs.
method Decomposed generalization error into approximation and training errors; derived gradient flow in the wide network limit.
result Generalization error tends to zero as the number of neurons and training time tend to infinity.
Gradient estimates for a parabolic PDE under Ricci-Bourguignon flow on warped product manifolds.
problem Analyzing the Ricci-Bourguignon flow on warped product manifolds.
method Establishing gradient estimates for a parabolic partial differential equation.
result Gradient estimates for the parabolic PDE provide analytic input for geometric applications.
This research analyzes deep PDE solvers for option pricing accuracy.
problem Understanding the accuracy of deep learning methods for solving PDEs in option pricing.
method Comparative experiments with two neural network algorithms in Black--Scholes and Heston models.
result Empirical convergence rates and training times of TDGF method determined.
The Sinkhorn flow converges to a Wasserstein mirror gradient flow from the Sinkhorn algorithm.
problem Optimizing joint distributions using the Sinkhorn algorithm.
method Wasserstein mirror gradient flow derived from the Sinkhorn algorithm.
result The Sinkhorn flow converges to a Wasserstein mirror gradient flow.
We illustrate the flow or wave character of the metrics and curvatures of evolving manifolds, introducing the Riemann flow and the Riemann wave via the bialternate product Riemannian metric. This kind of evolutions are new and very natural to understand certain flow or wave phenomena in the nature as well as the geomet…
We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the L2-space produces the same evolution as the gradient flow of the relative entropy in the L2-Wasserstein space. This means that the heat flow is well defined by either one of the two gradient fl…
Unified framework for forward and inverse PDE problems in multiphase media.
problem Non-differentiable inverse problems in discrete-valued material fields.
method GenPANIS: Latent-variable generative framework preserving discrete microstructures.
result Unified bidirectional inference with minimal labeled pairs and physics-aware decoder.
Study on symmetries in wide neural networks' dynamics without bias.
problem Understanding symmetries in the dynamics of wide two-layer neural networks.
method Analyzing symmetries in gradient flow on population risk for infinitely wide networks.
result Symmetries can simplify the dynamics of predictors and reduce the dimensionality of the problem.
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.
The paper proposes a new method to approximate Wasserstein-Fisher-Rao flows using Monte Carlo techniques.
problem Sampling from probability distributions and minimizing Kullback-Leibler divergence.
method Sequential Monte Carlo approximations of Wasserstein-Fisher-Rao gradient flows.
result The proposed method outperforms other Monte Carlo algorithms in certain conditions.
Paper analyzes inclusive KL inference using Wasserstein gradient flows.
problem Analyzing inclusive KL inference with mathematical tools.
method Gradient flows derived from PDE analysis.
result Unified view of existing sampling algorithms as inclusive-KL inference.
This paper analyzes convergence of large-scale Transformers with weight decay.
problem Understanding optimization guarantees in large-scale Transformer training.
method Construct mean-field limit, show gradient flow convergence to PDE, demonstrate global minimum consistency.
result Gradient flow reaches global minimum in large-scale Transformers with small weight decay.
The paper proves convergence of normalized Ricci flow on compact manifolds.
problem Convergence of normalized Ricci flow on compact manifolds.
method Gradient inequality of Łojasiewicz type to show convergence to steady-states.
result Convergence of normalized Ricci flow to steady-states on compact manifolds.
Self-test loss functions improve data-driven modeling of weak-form operators and gradient flows.
problem Challenges in selecting test functions for data-driven modeling involving weak-form operators and gradient flows.
method Introducing self-test loss functions that depend on unknown parameters and are quadratic.
result Self-test loss functions conserve energy for gradient flows and coincide with log-likelihood ratios for stochastic differential equations.
We consider applications of the theory of balanced weight filtrations and iterated logarithms, initiated in arXiv:1706.01073, to PDEs. The main result is a complete description of the asymptotics of the Yang--Mills flow on the space of metrics on a holomorphic bundle over a Riemann surface. A key ingredient in the argu…
Paper confirms Thom's conjecture for nonlinear evolutions on manifolds.
problem Thom's gradient conjecture for nonlinear evolution equations.
method Extending and settling the conjecture in infinite dimensional problems using Łojasiewicz, L. Simon, and Kurdyka-Mostowski-Parusinski's foundational works.
result Uniqueness of the limiting direction and characterization of convergence rates for both classical and infinite dimensional settings.
Toda flow explained as a porous medium equation.
problem Understanding the Toda flow through the lens of porous medium equations.
method Analyzing the geometry and dynamics of the porous medium equation and comparing it to the Toda flow.
result The Toda flow can be represented as a specific porous medium equation, revealing its gradient and Hamiltonian nature.
This paper analyzes deep and wide transformer training dynamics.
problem Understanding the training dynamics of infinitely deep and wide transformers.
method Develops a mean-field framework for gradient-based training of transformers, controlling a neural PDE.
result Establishes a rigorous foundation for gradient-based transformer training, proving convergence to global minima.
Method learns PDE dynamics via evolving latent manifold using Ricci flow.
problem Learning dynamics in time, especially PDEs, with low-dimensional representations.
method Parameterizes latent manifold, simulates Ricci flow physics-informedly, matching manifold quantities.
result Ricci flow facilitates learning for out-of-distribution data and adversarial robustness.
We develop VAE-DLM for dynamics with geometric flows in latent space.
problem Learning latent geometric properties for dynamics in high-dimensional data.
method Riemannian approaches to VAEs with a geometric flow in latent space, reformulating ELBO loss.
result Improved performance and robust learning for external dynamics, reducing OOD error.
Discussing curvature flows and their applications.
problem Analyzing expanding curvature flows.
method Classical aspects of expanding curvature flows.
result First applications of curvature flows.
Survey of geometric flows from unified string theories.
problem None explicitly stated, but related to understanding geometric flows in string theories.
method Survey of geometric flows in various geometries (complex, almost-complex, symplectic) motivated by string theories.
result Intermediate flows between Ricci and Kähler-Ricci flows, often coupled to additional fields.
A fundamental problem in Bayesian inference and statistical machine learning is to efficiently sample from multimodal distributions. Due to metastability, multimodal distributions are difficult to sample using standard Markov chain Monte Carlo methods. We propose a new sampling algorithm based on a birth-death mechanis…
Flow Annealing Posterior Sampling unifies stochastic-process regression and PDE inverse problems.
problem Function-space posterior sampling for stochastic processes and inverse problems.
method Flow Annealing Posterior Sampling (FAPS) using pretrained function-space flow-matching priors.
result Coherent posterior samples with accurate uncertainty quantification.
We consider some elementary aspects of the geometry of the space of probability measures endowed with Wasserstein distance. In such a setting, we discuss the various terms entering Perelman's shrinker entropy, and characterize two new monotonic functionals for the volume-normalized Ricci flow. One is obtained by a resc…
New method models PDEs from noisy, limited data.
problem Modeling PDEs with incomplete, noisy data.
method Learned linear transformation of spatial grid points, followed by dynamics learning in a reduced basis, then back transformation.
result Rapid high-resolution simulations with smaller training data sets.
New PDE systems generalize Hawking mass monotonicity.
problem Generalizing Hawking mass monotonicity to initial data sets.
method Introduced new systems of PDE on initial data sets (M,g,k). result Generalized Geroch's monotonicity formula to initial data sets.
FM4PDE learns PDE solutions from sparse data.
problem Reconstructing PDE solutions from limited observations.
method Flow-matching generative framework that learns PDE coefficients and solutions.
result Error guarantees for guided procedures, including deterministic and stochastic samplers.
Lecture notes on mean curvature flow for beginners.
problem Existence and uniqueness of mean curvature flow solutions.
method Conditional existence and weak-strong uniqueness theory.
result Accessible introduction for graduate students.
We give a twistorial interpretation of geometric structures on a Riemannian manifold, as sections of homogeneous fibre bundles, following an original insight by Wood (2003). The natural Dirichlet energy induces an abstract harmonicity condition, which gives rise to a geometric gradient flow. We establish a number of an…
In this note, we extend our previous work on the inverse σk problem. Inverse σk problem is a fully nonlinear geometric PDE on compact Kähler manifolds. Given a proper geometric condition, we prove that a large family of nonlinear geometric flows converges to the desired solution of the given PDE.
Develops a new framework to analyze gradient flow regimes and derive explicit solutions.
problem Analyzing scaling regimes and deriving explicit analytic solutions for gradient flow in large learning problems.
method Formal power series expansion of the loss evolution with coefficients encoded by diagrams.
result Reveals different learning phases and obtains explicit solutions in some cases.
New methods improve deep learning for solving linear PDEs.
problem Efficiently solving high-dimensional linear PDEs using deep learning.
method Rigorous investigation of gradient estimators for SDE-based variational formulations.
result Novel methods provide substantial performance improvements.
Deep ResNets can achieve zero loss with enough layers and weights.
problem Finding parameters in deep ResNets that fit training data perfectly.
method Mean-field analysis and gradient descent flow to a PDE.
result Gradient descent for ResNet training converges to a zero-loss solution in the large-NN limit.
PRISMA uses PDE residuals for fast, robust, and accurate inference.
problem Slow gradient-based optimization and instability in PDE residual-based methods.
method Integrates PDE residuals directly into the model's architecture via attention mechanisms in the spectral domain.
result Competitive accuracy with significantly lower inference costs and faster speeds.
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.
New method learns PDE solutions from low-fidelity data.
problem Challenges in learning PDE surrogates with scarce data.
method Flow matching in infinite-dimensional space with conditional neural operators.
result Accurately learns PDE solutions across different resolutions and fidelities.
Gradient descent finds global optima in ResNets with sufficient parameters.
problem Finding optimal parameters in ResNet models.
method Mean-field analysis and gradient-flow PDE to study convergence of first-order optimization methods.
result First-order methods can find global minimizers in overparameterized ResNets.
New method solves high-dimensional PDEs fast using physics-informed neural networks.
problem High computational cost in solving high-dimensional PDEs.
method Stochastic Dimension Gradient Descent (SDGD) for physics-informed neural networks (PINNs).
result Solves many high-dimensional PDEs including HJB and Schrödinger equations in 100,000 dimensions in 12 hours.
Solves the gauge problem for Ricci flow cylinders, proving strong rigidity.
problem Recognizing metrics in different coordinates and diffeomorphisms.
method Solves a nonlinear system of PDEs to produce a diffeomorphism fixing a gauge.
result Strong rigidity of cylinders in Ricci flow, proving all tangent flows are cylinders.
The paper shows how neural networks can approximate PDEs with polynomial scaling in dimension.
problem Understanding the complexity of approximating PDE solutions with neural networks.
method Developed a proof technique to simulate gradient descent using neural networks.
result Neural network parameters scale polynomially with input dimension for approximating PDE solutions.
A new machine learning method solves high-dimensional Kolmogorov PDEs efficiently.
problem Solving high-dimensional Kolmogorov PDEs and SDEs.
method Stochastic weighted minimization and stochastic gradient descent with Malliavin weights.
result Accurate approximation of high-dimensional Kolmogorov PDEs and SDEs without curse of dimensionality.
Enhances neural network solvers for PDEs with complex boundary conditions.
problem Challenges in solving PDEs with high accuracy and complex boundary conditions.
method Integrates natural gradient optimization with numerical time-stepping schemes to enforce Dirichlet boundary conditions.
result Superior accuracy and computational efficiency of the proposed methods for solving PDEs.
Meta-learning base distributions for efficient PDE solutions.
problem Efficiently solving parametric parabolic PDEs across different scenarios.
method Meta-learning base distributions to compute PDE solutions.
result Improves generalization to new parameter regimes.
Efficiently samples conformal boundaries in high dimensions using flows.
problem Difficulty in interpreting and using prediction sets in high-dimensional or structured output spaces.
method Flow-based approach using differentiable nonconformity scores to induce deterministic flows on the output space.
result Sampling conformal boundaries in arbitrary dimensions becomes computationally efficient and training-free.
The paper analyzes sensitivities of cash flows using PDEs and Hansen-Scheinkman decomposition.
problem Large-time sensitivities of cash flows in quantitative finance.
method PDE representation of pricing operator with Hansen-Scheinkman decomposition.
result Detailed convergence rates of sensitivities are provided.