New method learns soliton dynamics from scattering data without assuming known equations.
problem Deriving soliton dynamics from scattering data without prior knowledge.
method Combining IST with weak-form system identification for data-driven discovery.
result Effective soliton dynamics models derived from observed scattering data.
We review the spectral analysis and the time-dependent approach of scattering theory for manifolds with asymptotically cylindrical ends. For the spectral analysis, higher order resolvent estimates are obtained via Mourre theory for both short-range and long-range behaviors of the metric and the perturbation at infinity…
WideBNet learns inverse scattering from wide-band data efficiently and stably.
problem Learning the inverse scattering map from wide-band scattering data.
method Combines butterfly factorization, FFT, and deep learning.
result WideBNet requires fewer training points and has stable training dynamics.
A dataset of 10 molecule types for machine learning studies.
problem Lack of suitable datasets for machine learning in molecular imaging.
method Generated 2D cross-sectional projections of 10 molecule types from Molecular Dynamics trajectories.
result Benchmark dataset for machine learning, deep learning, and image processing in scattering, imaging, and microscopy.
Deep learning solves wave-based inverse problems, including super-resolution imaging.
problem Solving inverse wave scattering problems across all length scales.
method Wide-band butterfly network coupled with dynamic noise injection.
result Framework successfully solves super-resolution imaging problems.
We construct a large class of dynamical vacuum black hole spacetimes whose exterior geometry asymptotically settles down to a fixed Schwarzschild or Kerr metric. The construction proceeds by solving a backwards scattering problem for the Einstein vacuum equations with characteristic data prescribed on the event horizon…
FAT-GAN simulates electron-proton scattering without theoretical assumptions.
problem Efficiently training GANs to simulate complex particle distributions.
method Developed FAT-GAN using transformed and augmented features to improve GAN performance.
result FAT-GAN accurately reproduces electron momenta distributions in electron-proton scattering.
The accurate characterization of the business cycles in the nonlinear dynamic financial and economic systems in the time of globalization represents a formidable research problem. The central banks and other financial institutions make their decisions on the minimum capital requirements, countercyclical capital buffer …
The study counts geodesics on curved surfaces with specific intersections.
problem Counting geodesics with exact intersection numbers on curved surfaces.
method Introduced a dynamical scattering operator and used Pollicott-Ruelle resonances.
result Asymptotic growth of geodesics with prescribed intersections.
The moduli space of static finite energy solutions to Ward's integrable chiral model is the space MN of based rational maps from $\CP^1$ to itself with degree N. The Lagrangian of Ward's model gives rise to a Kähler metric and a magnetic vector potential on this space. However, the magnetic field strength vanishes…
New deep learning model estimates scattering timescale of FRBs efficiently.
problem Estimating scattering timescale of fast radio bursts (FRBs) is a bottleneck.
method Multimodal Transformer Based Generic Mixture Density Network (MT-GMDN) that ingests dynamic spectrum and timeseries profile.
result Achieves 94% R2 on expected value of τ for measurable scattering. Survey of spectral theory and dynamics for infinite volume hyperbolic manifolds.
problem Understanding infinite volume asymptotically hyperbolic manifolds.
method Survey of geometry, spectral theory, dynamics, and quantum/classical mechanics.
result Recent results, ideas, and conjectures discussed.
Vortex induced vibrations of bluff bodies occur when the vortex shedding frequency is close to the natural frequency of the structure. Of interest is the prediction of the lift and drag forces on the structure given some limited and scattered information on the velocity field. This is an inverse problem that is not str…
New MHSNs extract multiscale features from complex data for robust classification.
problem Signal classification and domain classification on complex data.
method Layered structure with multiscale basis dictionaries, pooling operations, and invariant features.
result High-accuracy classification with fewer parameters than traditional graph neural networks.
We introduce scattering-symplectic manifolds, manifolds with a type of minimally degenerate Poisson structure that is not too restrictive so as to have a large class of examples, yet restrictive enough for standard Poisson invariants to be computable. This paper will demonstrate the potential of the scattering symplect…
We introduce general scattering transforms as mathematical models of deep neural networks with l2 pooling. Scattering networks iteratively apply complex valued unitary operators, and the pooling is performed by a complex modulus. An expected scattering defines a contractive representation of a high-dimensional probabil…
Paper explains scattering diagrams' role in mirror symmetry.
problem Reconstruction problem in mirror symmetry.
method Introduction of scattering diagrams and their role in SYZ and HMS conjectures.
result Scattering diagrams help in understanding mirror symmetry.
New method uses broken scattering to uniquely identify Finsler manifolds.
problem Identifying Finsler manifolds from scattering data.
method Uses broken scattering relation to compare geodesics.
result Two reversible Finsler manifolds with the same broken scattering relation are isometric.
GSAN learns adaptive node representations using geometric scattering and attention.
problem Oversmoothing in node representation learning.
method Attention-based architecture integrating geometric scattering and GCN channels.
result GSAN outperforms previous networks in semi-supervised node classification.
Unified graph scattering transforms improve theoretical properties of graph neural networks.
problem Improving theoretical guarantees for graph neural networks.
method Introducing windowed and non-windowed geometric scattering transforms for graphs.
result Unified family of graph scattering transforms with provable stability and invariance.
Scattering representations simplify SBI for images without extra compression.
problem Efficiently performing simulation-based inference on images with limited data.
method Use scattering representations for compression and learning, combined with spatial averaging and expressive density estimators.
result Scattering representations provide more information than traditional methods, without requiring additional simulations.
The paper establishes scattering theory for wave equations on Schwarzschild spacetime.
problem Defocusing semilinear wave equations on Schwarzschild spacetime.
method Combining energy and pointwise decay results with Sobolev embedding, constructing scattering operator.
result Construction of a scattering operator mapping past to future scattering data.
Paper develops formulas for shape derivatives in wave scattering.
problem Computing high order shape derivatives for wave scattering is challenging.
method Introduces elegant recurrence formulas using differential forms and Lie derivatives.
result Unified framework for computing high order shape perturbations in scattering problems.
The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of the success of convolutional neural networks (ConvNets) in image data analysis and other tasks. Inspired by recent interest in geometric deep learning, which aims to generalize ConvNets to manifold and gra…
Study scattering rigidity on stationary manifolds using geodesics.
problem Scattering rigidity on standard stationary manifolds.
method Use Hamiltonian reduction to relate to MP-systems. result New rigidity results for stationary manifolds.
MODWST improves classification tasks with wavelet scattering.
problem Signal classification challenges.
method Combines MODWT and WST for feature extraction.
result MODWST outperforms CNNs in limited data scenarios.
Study on scattering geodesics on modular surface and their sojourn times.
problem Distribution of scattering geodesics and their sojourn times on modular surface.
method Analysis of scattering geodesics in modular surface, establishing connection to prime divisors in arithmetic progression.
result Established a connection between scattering geodesics and prime divisors in arithmetic progression.
Scattering theory on Riemann surfaces with explicit matrix and generalized period mappings.
problem Scattering theory for harmonic one-forms on Riemann surfaces.
method Construction of scattering theory from boundary value problems involving systems of curves and jump problems. Explicit expression for scattering matrix using Schiffer operators.
result Unitary scattering matrix and general association of polarizing Lagrangian spaces.
Study on recovering Lorentzian metrics from scattering data.
problem Recovering Lorentzian metrics from scattering data on a boundary.
method Analyzing the role of boundary distance functions and linearizing the light ray transform.
result Scattering rigidity can be reduced to boundary rigidity of magnetic systems.
The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric…
Bayesian Scattering offers a simple baseline for image data uncertainty.
problem Lack of interpretable, mathematically grounded uncertainty quantification methods for image data.
method Coupling wavelet scattering transform with a simple probabilistic head.
result Bayesian Scattering provides sensible uncertainty estimates under distribution shifts.
Scattering theory for linearised gravity on Schwarzschild black hole exterior.
problem Constructing a scattering theory for linearised gravity equations on Schwarzschild background.
method Building on previous work, constructing Hilbert space-isomorphisms for finite energy initial data and scattering states.
result Past and future linear memories are related by an antipodal map for Bondi-normalised solutions.
We construct continuous families of scattering manifolds with the same scattering phase. The manifolds are compactly supported metric perturbations of Euclidean Rn for n≥8. The metric perturbation may have arbitrarily small support.
Scattering theory for harmonic one-forms on Riemann surfaces.
problem Understanding harmonic one-forms on Riemann surfaces.
method Constructing scattering theory through boundary value problems and integral operators.
result Explicit expression for the scattering matrix and proof of unitarity.
Paper shows how scattering maps of Schrödinger equations relate to metrics.
problem Relating scattering maps of time-dependent Schrödinger equations to metrics.
method Analyzes scattering maps for specific classes of metrics and diffeomorphisms.
result Scattering maps differ by a compact operator if and only if metrics are related by diffeomorphism.
The problem of recovering the asymptotics of a short range perturbation of the Euclidean metric on R^n from fixed energy scattering data is studied. It is shown that if two such metrics, g1, g2, have scattering data at some fixed energy which are equal up to smoothing, then there exists a diffeomorphism ψ`fixing infini…
Scattering theory developed for linearised gravity near Schwarzschild black hole.
problem Linear stability of Schwarzschild spacetime and scattering of gravitational waves.
method Physical-space Chandrasekhar transformation and Teukolsky-Starobinsky correspondence.
result Construction of scattering theory for spin 2 Teukolsky equations.
Establishes scattering theory for de Sitter vacuum solutions in even dimensions.
problem Quantitative nonlinear scattering theory for asymptotically de Sitter vacuum solutions in even spatial dimensions.
method Geometric Littlewood-Paley decomposition of the solution, constructing the scattering map.
result Existence and uniqueness of scattering states, asymptotic completeness, and invertible scattering map with quantitative control.
We develop a geometric scattering theory for a geometrically finite group acting on (a vector bundle over) a symmetric space of negative curvature. In particular, we obtain the meromorphic continuation of Eisenstein series and scattering matrices and their functional equations.
We develop the scattering theory of general conformally compact metrics. For low frequencies, the domain of the scattering matrix is shown to be frequency dependent. In particular, generalized eigenfunctions exhibit L^2 decay in directions where the asymptotic curvature is sufficiently negative. The scattering matrix i…
Analytic metrics are uniquely determined by their scattering map.
problem Determining Riemannian manifolds from scattering data.
method Analytic negatively curved Riemannian manifolds with strictly convex boundary.
result The scattering map determines the manifold up to isometry.
Wavelet scattering spectra model non-Gaussian time-series, proving scale invariance for self-similar processes.
problem Modeling non-Gaussian time-series with stationary increments.
method Complex wavelet transform for scale variations, joint correlation matrix for scale dependencies, second wavelet transform for diagonalization, maximum entropy models conditioned by scattering spectra coefficients.
result Scattering spectra of self-similar processes are scale invariant, allowing statistical testing and generation of new time-series.
We prove that if two non-trapping obstacles in Rn satisfy some rather weak non-degeneracy conditions and the scattering rays in their exteriors have (almost) the same travelling times or (almost) the same scattering length spectrum, then they coincide.
Unified geometric scattering model for measure spaces.
problem Improving CNNs for non-Euclidean data.
method Unified geometric scattering model for measure spaces.
result Unified model includes previous work and applies to more general settings.
Inverse scattering result on AH manifolds determines metric up to diffeo and conformal factor.
problem Determining the metric on an AH manifold from scattering data.
method Relating eigenvalue problem to Conformal Laplacian and using Guillarmou--Guillopé and Chang--González results.
result Scattering matrix at energy 2n+1 determines jet of the metric on the boundary up to diffeomorphism and conformal factor. Scattering networks are a class of designed Convolutional Neural Networks (CNNs) with fixed weights. We argue they can serve as generic representations for modelling images. In particular, by working in scattering space, we achieve competitive results both for supervised and unsupervised learning tasks, while making pr…
Study of scattering on singular Yamabe spaces using asymptotically hyperbolic manifolds.
problem Understanding conformal geometry of compact manifolds with boundary.
method Application of scattering theory to singular Yamabe metrics.
result Definition of extrinsic GJMS operators and Q-curvatures on boundary.
Scattering transforms are non-trainable deep convolutional architectures that exploit the multi-scale resolution of a wavelet filter bank to obtain an appropriate representation of data. More importantly, they are proven invariant to translations, and stable to perturbations that are close to translations. This stabili…