Extracts coarse-grained PDEs from microscopic simulations.
problem Discovering effective PDEs for macro-scale processes from microscopic data.
method Combining neural networks with equation-free numerics and data-driven approaches.
result Efficiently discovers macro-scale PDEs from microscopic simulations.
Local Neural Operators enable efficient system-level analysis of complex PDEs.
problem System-level analysis of large-scale dynamical systems using neural operators.
method Integrating local Neural Operators with Krylov subspace iterative methods for stability and bifurcation analysis.
result Demonstrated effectiveness of local Neural Operators in fixed-point, stability, and bifurcation analysis of nonlinear PDEs.
Derives PDEs from data using manifold learning and neural networks.
problem Identifying PDEs from unknown variables and dynamics.
method Combines manifold learning (Diffusion Maps) and neural networks.
result Emergent space identification connects with multiscale computation.
Study proves all free boundary CMC annuli are of finite type.
problem Free boundary constant mean curvature annuli in the unit ball.
method Adapted Sklyanin's K-matrix formalism to sinh-Gordon equation.
result All free boundary CMC annuli are of finite type.
Smooth solutions up to evolving free boundaries for degenerate equations.
problem Degenerate parabolic equations with evolving free boundaries.
method Smooth short-time existence using linear degenerate equations on a fixed domain.
result Smoothness up to the free boundary for the p-Laplacian evolution equation and α-Gauss curvature flow. The wave equation (free boson) problem is studied from the viewpoint of the relations on the symplectic manifolds associated to the boundary induced by solutions. Unexpectedly there is still something to say on this simple, well-studied problem. In particular, boundaries which do not allow for a meaningful Hamiltonian …
Solves a general class of free boundary Monge-Ampère equations.
problem Optimal transport with degenerate densities and geometric problems.
method Analyzes a specific class of Monge-Ampère equations and their applications.
result Solves the equations for a general class, including applications to optimal transport and geometric problems.
Framework detects tipping points in complex systems using ML.
problem Detecting tipping points in complex, emergent systems.
method Combining manifold learning, neural networks, and Gaussian processes.
result Reduced-order models for mesoscopic and mean-field dynamics.
TURB-Rot provides a large database of turbulent rotating flow snapshots for research.
problem Lack of large-scale, high-resolution datasets for turbulent rotating flows.
method Direct Numerical Simulations of Navier-Stokes equations with rotation.
result Provides a diverse set of 300K complex images and fields for testing.
We demonstrate the application of an algorithmic trading strategy based upon the recently developed dynamic mode decomposition (DMD) on portfolios of financial data. The method is capable of characterizing complex dynamical systems, in this case financial market dynamics, in an equation-free manner by decomposing the s…
This paper analyzes optimal consumption strategies for loss-averse investors with multiplicative habit formation.
problem Optimal consumption strategies for loss-averse investors with multiplicative habit formation.
method The study uses a concave envelope of the S-shaped utility function and a nonlinear free boundary problem to analyze the HJB equation.
result The paper provides optimal consumption and investment policies in feedback form.
The inference of correlated signal fields with unknown correlation structures is of high scientific and technological relevance, but poses significant conceptual and numerical challenges. To address these, we develop the correlated signal inference (CSI) algorithm within information field theory (IFT) and discuss its n…
LSS learns molecular trajectories from MD data.
problem Limited integration time steps in MD simulations.
method Three deep learning networks for slow collective variables, dynamics, and configuration reconstruction.
result Generates ultra-long synthetic folding trajectories.