The paper derives the QGS equations using stochastic central extensions.
problem Deriving the viscous quasi-geostrophic equations on the torus.
method Central extensions of Lie groups and Lie algebras, stochastic Lagrangian formulation, and Euler-Poincaré reduction.
result Stochastic perturbations to the central extension lead to solutions of the QGS equations.
The paper examines curvature and stability in quasi-geostrophic motions using spherical harmonics.
problem Analyzing the curvature and stability of quasi-geostrophic motions.
method Utilizing spherical harmonics and structure constants, the curvature of the L2 metric on the central extension is computed. result A lower bound for weather prediction error in a simplified model is suggested.
Geodesic interpretation of global quasi-geostrophic equations on sphere.
problem Modeling dynamics on the sphere using geodesic equations.
method Interpreting equations as geodesic on central extension of quantomorphism group.
result Global-in-time existence and uniqueness of solutions with stabilizing Lamb parameter effect.
Study shows singular sets for certain fluid equations are negligible.
problem Understanding singular sets in fluid dynamics equations.
method Spectral analysis of divergence-free vector fields and operator properties.
result Singular sets are Gaussian null sets for two-dimensional equations.
We present a numerical model for the dynamics of thin viscous threads based on a discrete, Lagrangian formulation of the smooth equations. The model makes use of a condensed set of coordinates, called the centerline/spin representation: the kinematical constraints linking the centerline's tangent to the orientation of …
On curved spaces, viscous fluids reach equilibrium quickly.
problem Thermalization of viscous fluids on negatively curved manifolds.
method Stochastic Navier-Stokes equations with kinematically selected deformation Laplacian.
result Exponential thermalization rate of $2νλ_\Def$.
New model combines physics and machine learning for ocean dynamics.
problem Discovering hidden laws governing ocean dynamics.
method Develops Deep Neural Numerical Models (DNNMs) to learn hidden variables of physical laws.
result Illustrates DNNMs applied to Sea Surface Height dynamics, connecting to QG model.
Study the geometry of hydrodynamics equations using diffeomorphism groups.
problem Investigate the Euler equations and surface quasi-geostrophic equation family.
method Realize equations as geodesic equations on diffeomorphism groups and analyze Riemannian exponential maps.
result Show precise conditions for non-linear Fredholm maps of index 0.
New approach connects UQ in SciML to viscous HJ PDEs for efficient uncertainty quantification.
problem Challenges in interpretability and expensive training procedures in UQ for SciML.
method Established connection between Bayesian inference and viscous HJ PDEs, developed Riccati-based methodology.
result Efficiently updates model predictions without retraining or data access, suitable for real-time inferences.
Identifies conjugate points in spherical harmonics solutions of quasi-geostrophic equations.
problem Locating conjugate points in spherical harmonics solutions.
method Utilizing structure constants and quasi-geostrophic equations on the sphere, identifying conjugate points.
result Existence and location of conjugate points along spherical harmonics solutions.
SDA method reduces memory and time for assimilating noisy geophysical data.
problem Challenges in identifying state trajectories of high-dimensional geophysical systems.
method Score-based data assimilation with modified score network architecture.
result Promising results for a two-layer quasi-geostrophic model.
The paper analyzes thin-shell limits for viscous operators on Riemannian hypersurfaces.
problem Analyzing boundary conditions and thin-shell limits for viscous operators on arbitrary smooth hypersurfaces.
method Decomposing the ambient Bochner Laplacian into intrinsic and radial pieces, proving results for stress-free and Hodge boundary conditions.
result Universal thin-shell limits for viscous operators on arbitrary smooth hypersurfaces, including stress-free and Hodge boundary conditions.
In this paper, using Riemann-Lagrange geometrical methods, we construct a geometrical model on 1-jet spaces for the study of multi-time relativistic magnetized non-viscous plasma, characterized by a given energy-stress-momentum distinguished (d-) tensor. In that arena, we give the conservation laws and the continuity e…
Using a simple and well-motivated modification of the stress-energy tensor for a viscous fluid proposed by Lichnerowicz, we prove that Einstein's equations coupled to a relativistic version of the Navier-Stokes equations are well-posed in a suitable Gevrey class if the fluid is incompressible and irrotational. These la…
The goal of the present work is twofold. First we prove the existence of an Hilbert Manifold structure on the space of immersed oriented closed surfaces with three derivatives in L2 in an arbitrary sub-manifold Mm of an euclidian space RQ. Second, using this Hilbert manifold structure, we prove a lower semi co…
EnSF uses image inpainting to handle partial observations in data assimilation.
problem Data assimilation challenges with partial observations.
method EnSF integrates image inpainting with diffusion models to predict unobserved states.
result EnSF successfully tracks SQG dynamics with partial observations.
We report analytical results for the development of the viscous fingering instability in a cylindrical Hele-Shaw cell of radius a and thickness b. We derive a generalized version of Darcy's law in such cylindrical background, and find it recovers the usual Darcy's law for flow in flat, rectangular cells, with correctio…
Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.
problem Deriving new equations for magnetic systems and proving their well-posedness.
method Introducing the magnetic Euler-Arnold equation and proving well-posedness for specific equations.
result Local and global well-posedness results for the magnetic Euler-Arnold equation associated with the global quasi-geostrophic equations.
This paper is concerned with the following Markovian stochastic differential equation of mean-reversion type \[ dR_t= (θ+σα(R_t, t))R_t dt +σR_t dB_t \] with an initial value R0=r0∈R, where θ∈R and σ>0 are constants, and the mean correction function $α:\mathbb{R}\times[0,\infty)\to α(x,t)\…
We demonstrate that the surface quasi-geostrophic (SQG) equation given by θt+⟨u,∇θ⟩=0,θ=∇×(−Δ)−1/2u, is the geodesic equation on the group of volume-preserving diffeomorphisms of a Riemannian manifold M in the right-invariant H˙−1/2 metric. We show by exampl…
In this article we study the induced geodesic distance of fractional order Sobolev metrics on the groups of (volume preserving) diffeomorphisms and symplectomorphisms. The interest in these geometries is fueled by the observation that they allow for a geometric interpretation for prominent partial differential equation…
Machine learning improves model forecasts by correcting errors.
problem Improving short- to mid-range forecasts by correcting model errors.
method Iterative method combining data assimilation and machine learning.
result Hybrid models outperform original models in forecasts.
We study the problem of coupling Einstein's equations to a relativistic and physically well-motivated version of the Navier-Stokes equations. Under a natural evolution condition for the vorticity, we prove existence and uniqueness in a suitable Gevrey class if the fluid is incompressible, where this condition is given …
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.
In this paper, we compute the sectional curvature of the quantomorphism group Dq(M) whose geodesic equation is the quasi-geostrophic (QG) equation in geophysics and oceanography, for flows with a stream function depending on only one variable. Using this explicit formula, we will also derive a criterion fo…
PINNs struggle with data-to-PDE inconsistencies, limiting their accuracy.
problem Data inconsistency in PINNs affects their accuracy and convergence.
method Systematic analysis of PINNs with varying data fidelity and residual errors.
result PINNs saturate at an error level dictated by data inconsistency.
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…
It is shown phenomenologically that the fractional derivative ξ=Dαu of order α of a multifractal function has a power-law tail ∝∣ξ∣−p⋆ in its cumulative probability, for a suitable range of α's. The exponent is determined by the condition ζp⋆=αp⋆, where ζp is the exponent of…
The energy in a square membrane Ω subject to constant viscous damping on a subset ω⊂Ω decays exponentially in time as soon as ω satisfies a geometrical condition known as the "Bardos-Lebeau-Rauch" condition. The rate τ(ω) of this decay satisfies τ(ω)=2min(−μ(ω),g(ω)) (see Lebeau [Math. Phys. Stud. …
Paper presents a new approach to continuum mechanics using port-Hamiltonian framework.
problem Geometric formulation of solid and fluid mechanics.
method Port-Hamiltonian framework, Dirac structures, Hamiltonian reduction theory.
result Systematic derivation of port-Hamiltonian models for solid and fluid mechanics.
Develops a method for identifying structured dynamical systems from data.
problem Identifying structured dynamical systems from undersampled and noisy data.
method Sparse least-squares fitting via ℓ1−ℓ2 optimization with the alternating direction method of multipliers. result The method is stable and successful under certain conditions, as shown by theoretical guarantees and computational results.
Study on friction forces for nonholonomic systems using affine connections.
problem Realizing nonholonomic constraints with strong friction forces.
method Affine connection approach, covariant derivatives, recursive procedure.
result Approximations of slip velocities and dynamics up to second order.
Develops neural network approximations for infinite-dimensional input-output maps.
problem Approximating input-output maps between infinite-dimensional spaces.
method Combines neural networks and model reduction techniques.
result Proves convergence of the proposed approximation methodology.
Model reduction methods aim to describe complex dynamic phenomena using only relevant dynamical variables, decreasing computational cost, and potentially highlighting key dynamical mechanisms. In the absence of special dynamical features such as scale separation or symmetries, the time evolution of these variables typi…
A new shape space allows optimization of non-smooth shapes in fluid mechanics.
problem Optimizing non-smooth shapes in fluid mechanics.
method Constructing a product manifold to include piecewise-smooth shapes.
result Numerical results show applicability in minimizing viscous energy dissipation.
The paper calibrates geophysical predictions using marginal distributions and machine learning.
problem Sensitivity to initial conditions in geophysical systems leads to large deviations in long-term forecasts.
method The method introduces a calibration algorithm based on normalization and Kernelized Stein Discrepancy (KSD) to enhance ML predictions.
result The method improves the fidelity of ML predictions to known physical distributions, ensuring consistency with non-local statistical structures.
RandNet-Parareal uses neural networks to speed up time-parallel PDE solving.
problem Solving systems of time-dependent differential equations efficiently.
method Combines Parareal's sequential and parallel approach with random neural networks.
result Achieves up to 125x and 22x speedup compared to existing methods.
The paper introduces comprehensive quasi-Einstein spacetimes and explores their properties.
problem Exploring new types of spacetimes in general relativity.
method Mathematical analysis of geometric and physical properties of comprehensive quasi-Einstein manifolds.
result Existence of comprehensive quasi-Einstein spacetimes and their properties.
Paper introduces mcTangent for real-time dynamical systems.
problem Real-time accurate solutions for complex dynamical systems.
method Synergy of tangent slope learning, model-constrained approach, sequential learning, and data randomization.
result Robust and long-time accurate solutions for various equations.
A new method predicts non-Markovian closure terms for complex systems.
problem Predicting the effect of unresolved variables on resolved dynamics in high-dimensional systems.
method Mamba-Assisted Closure (MAC) framework: sequence model trained to predict closure from resolved trajectory, coupled with reduced-order equations.
result Substantially outperforms existing methods in predictive accuracy and long-time stability.
Deep neural network predicts multiphase flow in heterogeneous domains.
problem Predicting multiphase flow in complex, heterogeneous systems.
method Deep neural network model for handling permeability heterogeneity and learning interplay of forces.
result Highly accurate predictions of CO2 saturation distribution with computational efficiency.
This study evaluates the importance of design of experiments for PINN in physics-informed deep learning.
problem Accuracy of PINN predictions depends on the design of experiment scheme.
method Comparative study of five PDEs using different design of experiment schemes.
result Hammersley sampling-based PINN outperforms other design of experiment schemes.
Random feature model approximates PDE solutions efficiently.
problem Approximating solutions to PDEs with high-dimensional inputs and outputs.
method Random feature model applied to infinite-dimensional operators.
result Efficient and accurate approximation of PDE solutions.
Enhanced autoencoders improve ROMs for PDEs by capturing essential properties.
problem Autoencoders struggle to capture essential properties for accurate ROMs.
method Introduced symmetric Convolutional AutoEncoders (CAEs) that preserve manifold properties.
result Symmetric CAEs yield more accurate latent trajectories and robust models.
Improved model error correction online with neural networks in 4D-Var.
problem Reconstructing dynamics of imperfectly observed physical models.
method Weak-constraint 4D-Var framework with online neural network training.
result Online model error correction yields more accurate results than offline.
Survey and new results link hydrodynamics, molecular physics, and financial engineering.
problem Understanding financial engineering topics like Asian options and volatility swaps.
method Linking Kevin waves, Klein-Kramers, and Kolmogorov equations to financial models.
result Corrected the original solution of the Kolmogorov equation.
The paper analyzes the score field of diffusion models using Burgers dynamics.
problem Understanding the evolution of score fields in diffusion models.
method Analyzes the score field through Burgers-type evolution law for diffusion models.
result Identifies a universal \( anh\) interfacial term in the score field.
WSINDy algorithm proves robust to noise in identifying differential equations.
problem Identifying differential equations from noisy data.
method Weak-form sparse identification of nonlinear dynamics (WSINDy) algorithm.
result WSINDy is asymptotically consistent for a wide class of models, including Navier-Stokes and Kuramoto-Sivashinsky equations.