Unified framework for Arnold-type invariants via dual complexes and finite-difference structures.
problem Study of Arnold-type invariants of immersed curves and surfaces.
method Framework on dual complexes, locally normalized maps, finite-difference structures, and Shumakovitch-type identities.
result Unified evaluation of Arnold-type invariants St(1) and St(2) on dual skeleta. 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.
Kolmogorov-Arnold Networks achieve optimal convergence rates in nonparametric regression.
problem Nonparametric function approximation in multivariate settings.
method Structured additive and multiplicative KANs using B-splines.
result Achieve minimax-optimal convergence rate O(n−2r/(2r+1)) for Sobolev space functions. New integral expression quantizes Arnold strangeness.
problem Quantifying Arnold strangeness of plane curves.
method Integrating curvatures multiplied by densities, reformulating Arnold strangeness using Shumakovitch's partition function.
result Quantized Arnold strangeness includes rotation number and higher invariant terms.
In this article, we give proofs on the Arnold Lagrangian intersection conjecture on the cotangent bundles, Arnold-Givental Lagrangian intersection conjecture and the Arnold fixed point conjecture.
Develops methods for constructing exact, non-stationary solutions to Euler equations.
problem Constructing exact, non-stationary solutions to the incompressible Euler equations.
method Arnold's geometric framework with a generalized Coriolis force.
result Explicit, smooth, global-in-time solutions on curved surfaces and three-dimensional manifolds.
Kolmogorov-Arnold Networks improve deep learning adaptivity and can approximate Besov functions optimally.
problem Improving deep learning adaptivity and understanding approximation rates.
method Analyzing Besov norms and using Res-KANs for approximation.
result KANs can optimally approximate Besov functions at the optimal rate.
Proposes a new neural network architecture combining MLP and basis functions.
problem Function approximation and operator learning in scientific machine learning.
method Combines robust MLP inner functions with flexible basis functions outer functions.
result KKAN outperforms MLPs and KANs in function approximation and operator learning tasks.
Kolmogorov-Arnold Networks promise scalable performance in high dimensions.
problem Curse of dimensionality in multilayer perceptrons.
method Kolmogorov-Arnold representation theorem and interpolation methods.
result Kolmogorov-Arnold Networks achieve true freedom from the curse of dimensionality.
Paper develops a neural model to assess cascading extreme events.
problem Risk assessment of domino effects like earthquakes and tsunamis.
method Develops a Kolmogorov-Arnold neural network (KANE) framework.
result Estimates the probability of one extreme event triggering another.
New integer-valued functions for Legendrian knots.
problem Understanding Legendrian knots better.
method Using Legendrian fronts to derive integer-valued linear functions.
result Introduced new invariants similar to Arnold's basic invariant.
Explain Arnold's proof of the Morse index theorem using Maslov index.
problem Proving the Morse index theorem in Riemannian geometry.
method Using symplectic arguments and the Maslov index.
result Self-contained exposition of Arnold's proof.
Revisits the connection between neural networks and the Kolmogorov-Arnold theorem.
problem Explains the limitations of using the Kolmogorov-Arnold theorem to explain neural networks with multiple hidden layers.
method Derives modifications of the Kolmogorov-Arnold representation that transfer smoothness properties to the outer function and can be well approximated by ReLU networks.
result Shows that a deep neural network with most layers approximating the interior function is a more natural interpretation of the Kolmogorov-Arnold representation.
KANHedge improves hedging of high-dimensional options using learnable B-spline activation functions.
problem Challenges in high-dimensional option pricing and hedging due to the curse of dimensionality.
method Introduces KANHedge, a novel BSDE-based hedger leveraging Kolmogorov-Arnold Networks with learnable B-spline activation functions.
result KANHedge provides improved hedging performance, achieving significant reductions in hedging cost metrics.
We define the generalized connected sum for generic closed plane curves, generalizing the strange sum defined by Arnold, and completely describe how the Arnold invariants J± and St behave under the generalized connected sums.
Arnold-Liouville systems cannot be bi-Hamiltonian generically.
problem The bi-Hamiltonian structure of Arnold-Liouville systems.
method Proving that a specific class of smooth functions is a meagre subset for the Fréchet topology, which implies Arnold-Liouville systems cannot be bi-Hamiltonian.
result Generically, Arnold-Liouville systems cannot be bi-Hamiltonian.
The paper proves that most metrics satisfy a strong version of Arnold's conjecture for Laplace eigenvalues.
problem Understanding metrics that satisfy a strong version of Arnold's conjecture for Laplace eigenvalues.
method Using geometric characterizations and perturbation theory, the paper proves the conjecture for most metrics.
result The Strong Arnold Hypothesis is satisfied for all metrics except for a set of infinite codimension.
Solves an Arnold trivium problem using calculus and topology.
problem Finding critical points on a two-dimensional surface.
method Lagrange multipliers, Morse theory, Poincare-Hopf theorem.
result Determines the genus of a two-dimensional surface.
Proves Arnold-Thom conjecture for surfaces' arrival times.
problem Existence of limit tangents for gradient flow lines of surfaces.
method Gradient flow lines of mean curvature flows with neck or cylindrical singularities.
result Proves Arnold's conjecture for all mean convex mean curvature flows of surfaces.
Explains Arnold's J+ invariant for curves, using basic math.
problem Understanding Arnold's J+ invariant for planar curves.
method Explains computation methods like Viro's sum.
result Basic undergraduate math suffices to grasp the invariant.
Wav-KAN improves neural network interpretability and performance.
problem Challenges in interpretability, training speed, robustness, and performance of traditional neural networks.
method Integrates wavelet functions into the Kolmogorov-Arnold network structure for efficient data representation.
result Enhanced accuracy, faster training speeds, and increased robustness compared to existing methods.
A new Kolmogorov-Arnold network improves function approximation and optimization.
problem Approximating potentially irregular functions in high dimensions.
method Proposes a new Kolmogorov-Arnold network (KAN) and provides error bounds and universal approximation theorems.
result Outperforms multilayer perceptrons in accuracy and convergence speed for irregular functions.
Recently V. Arnold introduced Strangeness and J± invariants of generic immersions of an oriented circle to R2. Here these invariants are generalized to the case of generic immersions of an oriented circle to an arbitrary surface F. We explicitly describe all the invariants satisfying axioms, which naturall…
causalKANs provides interpretable treatment effect estimates using neural networks.
problem The opacity of deep neural networks limits their adoption in sensitive domains.
method Proposes causalKANs, a framework that transforms neural estimators into interpretable closed-form formulas.
result causalKANs performs on par with neural baselines in CATE error metrics and offers a favorable accuracy--interpretability trade-off.
GC-KAN uses KANs to detect Granger causality in time series data.
problem Detecting causal relationships in nonlinear time series data.
method Developed GC-KAN framework using Kolmogorov-Arnold networks for Granger causality detection.
result KANs outperform MLPs in identifying sparse Granger causal relationships.
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.
Proves Arnold conjecture for singular symplectic manifolds using novel techniques.
problem Hamiltonian dynamics on singular symplectic manifolds.
method Introducing smooth symplectic forms to singular symplectic structures under mild conditions, using Floer homology.
result Proves a lower bound on the number of 1-periodic Hamiltonian orbits for b2m-symplectic manifolds. Study shows almost all Arnold stable solutions have no conjugate points.
problem Existence of conjugate points in Arnold stable solutions.
method Analysis of Misiołek curvature for Arnold stable solutions.
result Almost all Misiołek curvature is nonpositive for Arnold stable solutions.
We apply Arnold's theory of generic smooth plane curves to Stark-Zeeman systems. This is a class of Hamiltonian dynamical systems that describes the dynamics of an electron in an external electric and magnetic field, and includes many systems from celestial mechanics. Based on Arnold's J+-invariant, we introduce inv…
SVGP KAN integrates sparse variational GP with KANs for scalable probabilistic inference.
problem Lack of probabilistic outputs in standard KANs and cubic scaling of Gaussian Process methods.
method Sparse Variational GP-KAN combines KAN topology with sparse variational inference and permutation-based importance analysis.
result Enables probabilistic KANs to handle larger datasets with linear computational complexity.
KaCGM models provide transparent causal inference from tabular data.
problem Limited auditability in deep causal models for tabular data.
method KaCGM uses Kolmogorov-Arnold Networks to parameterize structural equations, enabling direct inspection and visualization of causal mechanisms.
result KaCGM achieves competitive performance and interpretable causal effects in real-world applications.
The paper confirms Arnold's conjecture about hyperbolic polynomials.
problem The number of connected components of hyperbolic polynomials increases linearly with degree.
method Constructive proof using homotopy invariance of the index of a curve and properties of homogeneous polynomials.
result Exact number of connected components of Hyp(D) is determined and representatives for each component are provided. In [R2] and [RO] the Arnold conjecture for closed symplectic manifolds with trivial second homotopy group was proved. This proof used surgery and cobordism theory. Here we give a purely cohomological proof of this result.
Adaptive RBF-KAN improves KANs by dynamically adjusting kernel parameters.
problem Efficiently approximating multivariate functions using learnable univariate edge functions.
method Integrates LOOCV-based kernel scale estimation with adaptive kernel learning.
result Adaptive RBF-KAN outperforms fixed kernel KANs on various benchmark functions.
We prove the Arnold conjecture for closed symplectic manifolds with π2(M)=0 and $\cat M=\dim M$. Furthermore, we prove an analog of the Lusternik-Schnirelmann theorem for functions with ``generalized hyperbolicity'' property.
Paper proposes a hybrid MTL framework for improved stock market prediction accuracy.
problem Inaccurate stock market predictions due to financial data's complexities.
method Multi-layer hybrid MTL structure with Transformer, BiGRU, and KAN.
result Achieved low MAE (1.078), MAPE (0.012), and high R^2 (0.98) compared to other models.
Proves Weinstein's and Arnold's conjectures using contact instantons.
problem Proving Weinstein's and Arnold's conjectures in contact geometry.
method Existence of fundamental class in Legendrian contact instanton cohomology, evaluation transversality, and geometric construction of contactomorphisms.
result Proves Weinstein's and Arnold's conjectures in full generality.
Novel KAN-based autoencoder improves asset pricing models' accuracy and interpretability.
problem Improving asset pricing models' accuracy and interpretability.
method Kolmogorov-Arnold Networks (KANs) inspired autoencoder for latent factor exposures.
result Outperforms Multilayer Perceptrons in both accuracy and interpretability.
Extends Arnold's linking theory to higher dimensions and submanifolds.
problem Volume-preserving actions in higher dimensions and submanifolds.
method Generalization of V. Arnold's theory to Rk and Rℓ. result Extension of asymptotic linking to higher dimensions and submanifolds.
Paper develops invariants for spherical curves using chord diagrams.
problem Developing invariants for spherical curves under local moves.
method Using based chord diagrams and local moves from Reidemeister moves.
result Invariants include both classical and new spherical curve invariants.
The aim of this paper is to compare stratifications of moduli spaces given by group actions in the case of similarity of matrices introduced by Arnold and the author's stratification by projective orbifolds, and its relation to deformations o elements in the moduli space.
S2KAN integrates symbolic primitives into neural network activations for improved interpretability.
problem Training activations in KANs often lack symbolic fidelity, leading to unintelligible models.
method Softly Symbolified Kolmogorov-Arnold Networks (S2KAN) integrates symbolic primitives into training with learnable gates and a Minimum Description Length objective.
result S2KAN discovers interpretable forms when symbolic terms suffice, gracefully degrading to dense splines when necessary.
A 3D metric conformally related to Arnold cat fast dynamo metric: dsA2=e−λzdp2+eλzdq2+dz2 is shown to present a behaviour of non-dynamos where the magnetic field exponentially decay in time. The Riemann-Christoffel connection and Riemann curvature tensor for the Arnold and its conformal counter…
KANEL combines models for early hit enrichment in virtual screening.
problem Assessing model accuracy in chemical bioactivity predictions.
method Ensemble workflow using Kolmogorov-Arnold Networks (KANs) and other models.
result Improves early hit enrichment metrics like PPV@N.
A new machine learning framework reduces IoT data transfer by two orders of magnitude.
problem Reducing data transfer in IoT devices over wireless channels.
method Developed a machine learning framework for distributed functional compression over GMAC and AWGN channels.
result The framework reduces communication by two orders of magnitude compared to cloud-based methods.
Kolmogorov-Arnold Networks offer interpretable models for energy applications.
problem Lack of interpretability in modern machine learning methods for sensitive industries.
method Symbolic regression with Kolmogorov-Arnold Networks compared to traditional feedforward neural networks.
result Kolmogorov-Arnold Networks yield perfectly interpretable models and learn real, physical relations.
KANs replace fixed MLP weights with learnable edge functions, improving accuracy and interpretability.
problem Lack of interpretability and scalability in MLPs.
method KANs use learnable activation functions on edges instead of fixed weights, replacing weights with spline functions.
result KANs outperform MLPs in accuracy and interpretability with smaller models.
A new KAN variant uses sinusoidal activations to approximate functions.
problem Approximating multivariable functions using neural networks.
method Replacing inner and outer functions in Kolmogorov-Arnold representation with weighted sinusoidal functions.
result The new KAN variant outperforms fixed-frequency Fourier transform and achieves comparable performance to MLPs.