Kolmogorov Flow

Kolmogorov flow, a fundamental model in fluid dynamics, studies the behavior of fluid flows under specific forcing conditions, often focusing on the transition to turbulence. Current research emphasizes developing efficient numerical methods, including deep neural networks and transformer architectures, to solve the associated partial differential equations and reconstruct flow fields from sparse, noisy data. These advancements improve the accuracy and scalability of simulations, enabling better understanding of complex flow dynamics and facilitating applications in areas like flow control and prediction. The development of robust and efficient algorithms for solving Kolmogorov equations is crucial for advancing our understanding of turbulence and other complex systems.

Papers

September 30, 2024

Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in $L^p$-sense
Ariel Neufeld, Tuan Anh Nguyen
Neural Network Deep Neural Network \Ell_p$ Norm Parabolic Partial Differential Equation Leaky ReLU Kolmogorov Flow

August 30, 2024

Reconstructing unsteady flows from sparse, noisy measurements with a physics-constrained convolutional neural network
Yaxin Mo, Luca Magri
Flow Field General Flow Noisy Measurement Physic Constrained Flow Reconstruction Kolmogorov Flow

December 22, 2023

Sharp error estimates for target measure diffusion maps with applications to the committor problem
Shashank Sule, Luke Evans, Maria Cameron
Financial Application Langevin Dynamic Diffusion Map Kolmogorov Flow Committor Function

December 15, 2023

Building symmetries into data-driven manifold dynamics models for complex flows
Carlos E. Pérez De Jesús, Alec J. Linot, Michael D. Graham
Long Term Manifold Learning Approximate Symmetry Navier Stokes Symmetry Breaking Invariant Manifold Kolmogorov Flow

September 24, 2023

Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the $L^p$-sense
Julia Ackermann, Arnulf Jentzen, Thomas Kruse, Benno Kuckuck, Joshua Lee Padgett
Deep Neural Network High Dimensional Partial Differential Equation \Ell_p$ Norm Leaky ReLU Canonical Lipschitz Mechanism Kolmogorov Flow

May 27, 2023

Scalable Transformer for PDE Surrogate Modeling
Zijie Li, Dule Shu, Amir Barati Farimani
Vision Transformer Transformer Based Surrogate Model Surrogate Modeling Projection Operator Kolmogorov Flow Efficient Surrogate Model

October 29, 2022

Data-driven low-dimensional dynamic model of Kolmogorov flow
Carlos E. Pérez De Jesús, Michael D. Graham
Data Driven Reduced Order Pattern Formation Low Dimensional Flow Dynamic Kolmogorov Flow Phase Dynamic

July 11, 2022

On uniform-in-time diffusion approximation for stochastic gradient descent
Lei Li, Yuliang Wang
Stochastic Gradient Descent Kolmogorov Flow

June 21, 2022

Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning
Lorenz Richter, Julius Berner
Deep Learning Robust Version Stochastic Differential Equation Variational Formulation Kolmogorov Flow Linear Partial Differential Equation

June 7, 2022

Computational Doob's h-transforms for Online Filtering of Discretely Observed Diffusions
Nicolas Chopin, Andras Fulop, Jeremy Heng, Alexandre H. Thiery
Computation Method Particle Filter Online Filtering Discrete Diffusion Kolmogorov Flow Nonlinear Diffusion

Kolmogorov Flow

Papers

Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in $L^p$-sense

Reconstructing unsteady flows from sparse, noisy measurements with a physics-constrained convolutional neural network

Sharp error estimates for target measure diffusion maps with applications to the committor problem

Building symmetries into data-driven manifold dynamics models for complex flows

Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the $L^p$-sense

Scalable Transformer for PDE Surrogate Modeling

Data-driven low-dimensional dynamic model of Kolmogorov flow

On uniform-in-time diffusion approximation for stochastic gradient descent

Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning

Computational Doob's h-transforms for Online Filtering of Discretely Observed Diffusions