Study shows how noise can ensure solutions to fluid dynamics equations.
problem Ensuring unique solutions to stochastic fluid dynamics equations.
method Extended existing results to linear advection of k-forms, proving existence and uniqueness of weak L^p-solutions.
result Proved existence and uniqueness of weak L^p-solutions to stochastic linear advection equation of k-forms.
Paper uses neural networks to speed up simulations of complex systems.
problem Rapid simulations of advection-dominated problems in engineering and geophysics.
method Recurrent neural network for approximating nonlinear component of ROM.
result The proposed framework accurately recovers transient dynamics without full nonlinear computations.
Proposes a new model for traffic flow on directed graphs.
problem Modeling advection on directed graphs for traffic flow.
method Reformulates graph advection operator as finite difference scheme; proposes DGAMGP model.
result Effective modeling of traffic flow and uncertainty as an advective process.
This work analyzes PINNs for advection-diffusion equations using NTK theory.
problem Understanding and resolving the training difficulties of PINNs for advection-diffusion equations.
method Neural Tangent Kernel (NTK) analysis of PINNs for the linear advection-diffusion equation (LAD).
result PINNs struggle due to spectral bias and convergence rate disparity, especially in advection-dominated and diffusion-dominated regimes.
Modeling bone microarchitecture adaptation using geometric flows.
problem Bone microarchitecture adaptation modeling.
method Advection and mean curvature flow model with a sphere as a test case.
result Closed-form solution for sphere under advection and mean curvature flow.
We discuss in this note applications of the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) to numerical solutions of partial differential equations arising from stochastic models in quantitative finance. In particular, we develop a framework for solving Black-Scholes-type equations by first t…
Paper tackles estimating initial conditions of spatio-temporal processes from sparse data.
problem Estimating initial conditions of spatio-temporal advection-diffusion processes from sparse data.
method Regularized convex optimization problem with Alternating Direction Method of Multipliers.
result Efficient solutions for non-uniform and shifted uniform sampling schemes.
We study the Euler-Lagrange equations for a parameter dependent G-invariant Lagrangian on a homogeneous G-space. We consider the pullback of the parameter dependent Lagrangian to the Lie group G, emphasizing the special invariance properties of the associated Euler-Poincaré equations with advected parameters.
We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection-diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or h…
The identification of sources of advection-diffusion transport is based usually on solving complex ill-posed inverse models against the available state- variable data records. However, if there are several sources with different locations and strengths, the data records represent mixtures rather than the separate influ…
Investigates fluid flow perturbations using geometric theory.
problem Analyzing linear perturbations in non-equilibrium fluid flows.
method Uses second order variations of the action and Jacobi fields.
result Demonstrates numerical simulations of perturbation dynamics.
We propose a method to compute optimal control paths for autonomous vehicles deployed for the purpose of inferring a velocity field. In addition to being advected by the flow, the vehicles are able to effect a fixed relative speed with arbitrary control over direction. It is this direction that is used as the basis for…
Cauchy invariants are now viewed as a powerful tool for investigating the Lagrangian structure of three-dimensional (3D) ideal flow (Frisch & Zheligovsky, Commun. Math. Phys., vol. 326, 2014, pp. 499-505, Podvigina et al., J. Comput. Phys., vol. 306, 2016, pp. 320-342). Looking at such invariants with the modern tools …
DeepMIDE forecasts wind speeds across space, time, and height for offshore wind energy.
problem Forecasting wind speeds across multiple heights for large offshore wind turbines.
method Statistical deep learning model that jointly models wind speeds at different heights using a multi-output integro-difference equation.
result DeepMIDE forecasts outperform traditional methods in real-world offshore wind energy data.
Robust PDE method for path-dependent Asian-style options using MPDATA.
problem Valuation of path-dependent Asian-style options.
method Non-oscillatory forward-in-time second-order MPDATA finite-difference scheme for solving 2D PDEs.
result MPDATA scheme improves solution over first-order upwind step, highlighting its importance.
New method converts video of dye plumes into PDEs for better understanding.
problem Inferring continuum models from uncalibrated video data.
method Develops a pipeline to convert grayscale recordings into scalar fields, isolates drift, and identifies transport laws.
result Selected reduced model outperforms advection-diffusion baselines and retains structural interpretability.
Study methods to recover unknown processes in PDEs from data.
problem Identifying unknown processes in time-dependent PDEs using observational data.
method Theoretical analysis and numerical approaches including Galerkin and collocation algorithms.
result The Galerkin algorithm is more suitable for practical situations with noisy data.
This paper introduces a linear state-space model with time-varying dynamics. The time dependency is obtained by forming the state dynamics matrix as a time-varying linear combination of a set of matrices. The time dependency of the weights in the linear combination is modelled by another linear Gaussian dynamical model…
Graph neural controlled differential equations learn graph dynamics from vertex observations.
problem Predicting future states of dynamical systems on graphs with limited vertex data.
method Incorporates graph topology information into NCDE to predict graph dynamics.
result Informed NCDE requires fewer parameters and lower MAE compared to previous methods.
New method uses PINNs to solve complex PDEs with sparse measurements.
problem Joint estimation of source and parameters in advection-diffusion equations with limited data.
method Weighted adaptive approach based on neural tangent kernel of PINNs.
result Successful estimation of source function, velocity, and diffusion parameters.
Study shows how heat leaks from material sets in low diffusivity scenarios.
problem Understanding heat leakage from material sets in low diffusivity limits.
method Generalized leading-order asymptotics for time-dependent diffusion processes.
result Diffusive transport out of a material set is proportional to the surface area of the set boundary.
Introduces numerical Gaussian process Kalman filtering for infinite-dimensional systems.
problem Kalman filtering on infinite-dimensional systems.
method Embedding numerical Gaussian processes into Kalman filter equations.
result Ability to perform Kalman filtering on infinite-dimensional systems using Gaussian processes.
We consider the concept of Stokes-Dirac structures in boundary control theory proposed by van der Schaft and Maschke. We introduce Poisson reduction in this context and show how Stokes-Dirac structures can be derived through symmetry reduction from a canonical Dirac structure on the unreduced phase space. In this way, …
SVGP KAN integrates uncertainty quantification into Kolmogorov-Arnold networks.
problem Uncertainty quantification in scientific machine learning models.
method Sparse variational Gaussian process inference with Kolmogorov-Arnold topology.
result Demonstrated ability to distinguish aleatoric and epistemic uncertainty in various scientific applications.
FiniteNet uses a neural network to improve PDE solving methods.
problem Improving accuracy in solving time-dependent PDEs.
method Fully convolutional LSTM network trained on simulation data.
result Reduces error by a factor of 2 to 3 compared to baseline methods.
Clarifies relation for solving control-affine Schrödinger bridge problems.
problem Solving control-affine Schrödinger bridge problems via Hopf-Cole transform.
method Applies Hopf-Cole transform to conditions of optimality, resulting in nonlinear PDEs.
result Generic control-affine Schrödinger bridge requires further algorithmic development.
Noise stabilizes solutions to transport equations, preventing blow-up.
problem Proving global existence and uniqueness of solutions to stochastic transport equations.
method Characteristics-based techniques exploiting the geometric structure of transport equations.
result Noise prevents blow-up in deterministic solutions and ensures global existence and uniqueness of solutions.
We study polygonal analogues of several moving boundary problems and their time discretization which preserves the constant area speed property. We establish various polygonal analogues of geometric formulas for moving boundaries and make use of the geometric formulas for our numerical scheme and its analysis of genera…
New method reduces PDE model parameters by 30% with sparsity.
problem Redundant parameters in neural network projections.
method Bregman iterations for sparsity, POD compression, bias propagation.
result 30% fewer parameters with similar accuracy.
Modeling wildfire aerosols using satellite data to predict solar radiation reduction.
problem Accurately estimate and predict AOD propagation from wildfires using multi-source satellite data.
method Physics-informed statistical modeling integrating multi-source satellite data with an advection-diffusion equation.
result The proposed approach accurately predicts AOD propagation and demonstrates model interpretability.
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.
Deep learning improves model discovery from sparse sensor data.
problem Improving physical understanding and predictions from coarse, non-grid sampled data.
method Physics-informed neural networks and automatic differentiation.
result Deep learning can recover underlying equations from sparse, non-grid data.
The purpose of this paper is to derive the anisotropic averaged Euler equations and to study their geometric and analytic properties. These new equations involve the evolution of a mean velocity field and an advected symmetric tensor that captures the fluctuation effects. Besides the derivation of these equations, the …
INDEQS: A Graph-Based Neural Controlled Differential Equation Framework for Forecasting
problem Forecasting time series with neural networks
method Incorporating prior knowledge of a directed graph
result Outer informedness consistently improves forecasting accuracy
A new algorithm computes Wasserstein barycenters without entropic regularization.
problem Computing Wasserstein barycenters efficiently and accurately.
method Free-support algorithm based on particle flow and Riemannian geometry.
result The algorithm avoids entropic regularization and is computationally tractable.
New guarantees for uniquely identifying transport maps and vector fields from finite measure-valued data.
problem Unique recovery of transport maps and vector fields from finite measure-valued data.
method Use of Whitney and Takens embedding theorems to establish conditions for unique identification.
result New metric for comparing diffeomorphisms and analogous results in infinitesimal settings.
USD algorithm transports distributions with or without mass conservation.
problem Transporting distributions with different masses.
method Particle descent algorithm using Sobolev-Fisher discrepancy.
result USD converges to target distribution in MMD sense.
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.
A contour integral method recently proposed by Weideman [IMA J. Numer. Anal., to appear] for integrating semi-discrete advection-diffusion PDEs, is extended for application to some of the important equations of mathematical finance. Using estimates for the numerical range of the spatial operator, optimal contour parame…
New algorithm speeds up Bayesian UQ for high-dimensional inverse problems.
problem Computational inefficiency in Bayesian inference for high-dimensional inverse problems.
method Deep neural network-based autoencoder for dimension reduction and emulation phase.
result Computational efficiency up to three orders of magnitude with scalable Bayesian UQ.
Study mass transport in low-diffusivity using Lagrangian coordinates.
problem Mass preserving transport of passive tracers in low-diffusivity limit.
method Lagrangian coordinates, time-averaged diffusion equation, weighted manifold structure.
result Leading order asymptotics extend to dominant nontrivial singular value in low-diffusivity limit.
Physics-guided reinforcement learning optimizes swimming in turbulent flows.
problem Optimizing swimming efforts to maintain proximity in turbulent environments.
method Physics-informed actor-physicist reinforcement learning algorithm.
result Physics-informed reinforcement learning outperforms standard methods in turbulent flow control.
We present a frame-invariant method for detecting coherent structures from Lagrangian flow trajectories that can be sparse in number, as is the case in many fluid mechanics applications of practical interest. The method, based on principles used in graph coloring and spectral graph drawing algorithms, examines a measur…
Adversarial reinforcement learning optimizes microswimmers' path-planning in turbulent flows.
problem Optimizing microswimmers' paths in turbulent flows for efficient target reach.
method Adversarial-reinforcement learning scheme applied to 2D and 3D turbulent flows.
result Microswimmers can reach targets faster than a naive approach in turbulent flows.
LGAC enhances heat transfer in turbulent boundary layers using slot jets.
problem Enhancing convective heat transfer in turbulent boundary layers.
method Artificial intelligence-based linear genetic algorithms control (LGAC) with slot jets.
result LGAC optimizes heat transfer and flow asymmetry in turbulent boundary layers.
Improved solver maintains positivity and accuracy across all time steps.
problem Linear second-order schemes for Fokker-Planck equation cannot preserve positivity.
method Flux-Corrected Diagonal Frog (FCDF) framework using nonlinear extension and iterative limiter.
result FCDF schemes are unconditionally positive across all time steps and maintain second-order accuracy.
Unified geometric formulation of Maxwell-Vlasov system using presymplectic and symmetry reduction.
problem Unified geometric formulation of Maxwell-Vlasov system.
method Skinner-Rusk formalism, presymplectic geometry, reduction by diffeomorphism group, affine Hamiltonian controls.
result Unified geometric structure unifying Lagrangian, Hamiltonian, gauge, reduction, and control-theoretic aspects.
In this paper, we consider the use of structure learning methods for probabilistic graphical models to identify statistical dependencies in high-dimensional physical processes. Such processes are often synthetically characterized using PDEs (partial differential equations) and are observed in a variety of natural pheno…