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.
Kolmogorov neural networks can represent various types of functions.
problem Representing different types of functions with neural networks.
method Continuous, discontinuous bounded or unbounded activation functions in a two hidden layer model.
result Kolmogorov neural networks can represent continuous, discontinuous bounded and all unbounded multivariate functions.
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.
Researchers use estimated Kolmogorov complexity for better link prediction in graphs.
problem Improving link prediction accuracy in complex networks.
method Regularization based on an approximation of Kolmogorov complexity, which is differentiable and compatible with recent link prediction algorithms.
result The regularization method shows good performance on diverse real-world networks, but the success is likely due to an aggregation method rather than actual estimation of Kolmogorov complexity.
Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov PDEs, respectively, are highly employed in models for the approximative pricing of …
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.
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.
Efficient algorithm approximates discrete random variables with minimal Kolmogorov distance.
problem Estimating the probability of missing deadlines in series-parallel schedules.
method An efficient algorithm that computes a random variable with minimal Kolmogorov distance to a given discrete random variable.
result The algorithm efficiently approximates the probability of missing deadlines with minimal Kolmogorov distance.
The usual derivation of the Fokker-Planck partial differential eqn. assumes the Chapman-Kolmogorov equation for a Markov process. Starting instead with an Ito stochastic differential equation we argue that finitely many states of memory are allowed in Kolmogorov's two pdes, K1 (the backward time pde) and K2 (the Fokker…
We study the solution to Kolmogorov-Feller equation and by using it provide pricing formulas of well known some options under jump-diffusion model.
Study finds conjugate points in geodesics of Kolmogorov flows on torus.
problem Characterizing pairs of integers (m,n) for which geodesics have conjugate points.
method Analysis of geodesics in the group of volume-preserving diffeomorphisms of a torus using stream functions.
result Existence of conjugate points for all pairs of strictly positive integers (m,n).
A new machine learning method solves high-dimensional Kolmogorov PDEs efficiently.
problem Solving high-dimensional Kolmogorov PDEs and SDEs.
method Stochastic weighted minimization and stochastic gradient descent with Malliavin weights.
result Accurate approximation of high-dimensional Kolmogorov PDEs and SDEs without curse of dimensionality.
Smooth KANs improve model reliability in computational biomedicine.
problem Limited convergence of KANs in representing generic smooth functions.
method Introducing smooth, structurally informed KANs that can approximate MLPs in specific function classes.
result Smooth KANs can achieve equivalence to MLPs in specific function classes, enhancing model reliability and performance.
K-DAREK improves KKANs for efficient function approximation with robust error bounds.
problem Efficient function approximation with uncertainty quantification for large-scale problems.
method Developed a novel learning algorithm, K-DAREK, for KKANs.
result Established robust error bounds that are distance-aware, improving efficiency and scalability.
We study gradient bounds and other functional inequalities for the diffusion semigroup generated by Kolmogorov type operators. The focus is on two different methods: coupling techniques and generalized Γ-calculus techniques. The advantages and drawbacks of each of these methods are discussed.
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. KSGAN uses KS distance for deep generative modeling.
problem Deep generative modeling challenges, especially for multivariate distributions.
method Formulates adversarial training as minimization of KS distance, using quantile function as critic.
result KSGAN trained distributions closely match target distributions.
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.
We consider several ways to measure the `geometric complexity' of an embedding from a simplicial complex into Euclidean space. One of these is a version of `thickness', based on a paper of Kolmogorov and Barzdin. We prove inequalities relating the thickness and the number of simplices in the simplicial complex, general…
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.
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.
NN-Turb generates turbulent velocity statistics using neural networks.
problem Creating a 1D field with turbulent velocity statistics.
method Fully-convolutional neural network (NN-Turb) to generate the field.
result NN-Turb generates a 1D field that satisfies Kolmogorov's 2/3 and 4/5 laws, exhibiting intermittency.
We link SVEs to SPDEs and derive Kolmogorov equations for singular kernels.
problem Solving stochastic Volterra equations with singular kernels.
method Establishing connections between SVEs and SPDEs, using stochastic calculus in Hilbert spaces.
result Solutions of SVEs can be expressed in terms of backward Kolmogorov equations.
Deep learning solves high-dimensional PDEs efficiently.
problem Solving high-dimensional Kolmogorov PDEs numerically.
method Reformulating as a statistical learning problem using Feynman-Kac formula.
result Single neural network learns entire family of PDEs.
We provide the proof that the space of time series data is a Kolmogorov space with T0-separation axiom using the loop space of time series data. In our approach we define a cyclic coordinate of intrinsic time scale of time series data after empirical mode decomposition. A spinor field of time series data comes fro…
The goal of this article is to describe the concepts of system dynamics and its applications to the simulation modeling of financial institutions daily activity. The hybrid method of the re-engineering of banking business processes based upon combination of system dynamics, queuing theory and tools of ordinary differen…
Poor approximators found in neural networks and random feature models.
problem Understanding why certain neural networks and models perform poorly in approximating functions.
method Established a scale separation of Kolmogorov width type and applied it to neural networks and random feature models.
result Reproducing kernel Hilbert spaces and two-layer neural networks are poor L2-approximators for certain functions. 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.
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.
Kolmogorov-Arnold Networks enable ultrafast online learning with fixed-point quantization.
problem Efficient online learning for high-frequency systems with strict memory constraints.
method Fixed-point online training on FPGAs exploiting B-spline locality in KANs.
result Kolmogorov-Arnold Networks are more efficient and expressive than MLPs for low-latency tasks.
Presented are two neural network architectures for convex functions, demonstrating competitive performance.
problem Approximating convex functions efficiently and accurately.
method Developed two neural network architectures: one based on linear-by-part representation and the other on cubic splines.
result Cubic ICKAN networks produce results similar to classical ICNNs in solving convex approximation problems.
The study improves volatility model pricing accuracy with new statistical expansions.
problem Improving option pricing accuracy in volatility models.
method Developed Edgeworth expansions for various volatility models.
result Enhanced statistical expansions for volatility models.
SurvLIME-KS improves survival model explanations robustly.
problem Improving explanations of unreliable survival models.
method SurvLIME-KS combines Cox proportional hazards model and Kolmogorov-Smirnov bounds for robust optimization.
result SurvLIME-KS minimizes average distance and maximizes distance in approximating cumulative hazard functions.
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.
KANOP uses KANs to efficiently price American options.
problem Efficiently pricing American options with limited data.
method Combines KANs with LSMC to estimate continuation value.
result KANOP provides more accurate option value estimates.
Kolmogorov-Arnold network improves GW catalog posterior construction.
problem Efficiently constructing posterior distributions for GW catalogs.
method Using the Kolmogorov-Arnold network to create lightweight neural density estimators.
result Kolmogorov-Arnold network achieves superior interpretability and accuracy in posterior construction.
New method speeds up SDE inference by matching moments to FPK equation.
problem Efficiency of sampling schemes in high-dimensional SDEs.
method Direct approximation of Fokker-Planck-Kolmogorov equation by matching moments.
result Fast, scalable inference in high-dimensional latent spaces.
New architectures improve KANs, making them more interpretable and accurate.
problem Improving Kolmogorov-Arnold networks while maintaining interpretability.
method Overprovisioned architectures combined with sparsification, deep supervision, and depth selection, optimized with a minimum description length objective.
result Combining sparsification with depth selection achieves competitive or superior accuracy while discovering smaller models.
HaKAN uses Hahn-KAN blocks to forecast multivariate time series.
problem Long-term time series forecasting challenges with high complexity and spectral bias.
method HaKAN integrates channel independence, patching, and a stack of Hahn-KAN blocks with residual connections. It uses Hahn polynomial-based learnable activation functions.
result HaKAN consistently outperforms state-of-the-art methods on various forecasting benchmarks.
Recent developments have linked causal inference with Algorithmic Information Theory, and methods have been developed that utilize Conditional Kolmogorov Complexity to determine causation between two random variables. We present a method for inferring causal direction between continuous variables by using an MDL Binnin…
A new feature selection method using random forest and Kolmogorov filter.
problem Ultra-high dimensional data feature selection.
method Fused Kolmogorov filter with random forest based recursive feature elimination.
result Selection and L2 consistency under weak conditions. We summarize our recent findings, where we proposed a framework for learning a Kolmogorov model, for a collection of binary random variables. More specifically, we derive conditions that link outcomes of specific random variables, and extract valuable relations from the data. We also propose an algorithm for computing …
Projects Markovian processes from Itô semimartingales with jumps.
problem Modeling Itô semimartingales with jumps using Markovian projections.
method Construct Markovian projections for Itô semimartingales with jumps using non-local FPKEs.
result Markovian projections match the marginal laws of the original process.
Accurate goodness-of-fit tests for the extreme tails of empirical distributions is a very important issue, relevant in many contexts, including geophysics, insurance, and finance. We have derived exact asymptotic results for a generalization of the large-sample Kolmogorov-Smirnov test, well suited to testing these extr…
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.
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.