Neural dynamical systems are dynamical systems that are described at least in part by neural networks. The class of continuous-time neural dynamical systems must, however, be numerically integrated for simulation and learning. Here, we present a compact neural circuit for two common numerical integrators: the explicit …
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48 results for “numerical continuation”
xVal tokenizes numbers continuously for better scientific model training.
problem Lack of continuous numerical tokenization for scientific datasets in LLMs.
method xVal: Continuous numerical tokenization strategy.
result xVal outperforms other numerical tokenization methods on scientific datasets.
Optimal binning method for numeric targets using mathematical programming.
problem Optimizing the discretization of numeric variables for classification.
method Mathematical programming formulation for binary, continuous, and multi-class targets with constraints.
result Convex mixed-integer programming formulations for all target types.
Proves solution uniqueness for biomembrane shape prediction.
problem Proving solution uniqueness for the genus one Canham variational problem.
method Combining numeric analytic continuation and singularity analysis to prove non-negativity of a sequence.
result Proves positivity of the sequence, leading to solution uniqueness.
Probabilistic Numeric SMC Sampling for Bayesian Nonlinear System Identification in Continuous Timestat.ML
This paper tackles Bayesian system identification with probabilistic numerical methods.
problem Accurately modeling nonlinear dynamic systems from noisy data.
method Probabilistic Sequential Monte Carlo (SMC) combined with probabilistic numerical integration.
result Efficient identification of latent states and system parameters from noisy measurements.
Binary encoding enables neural networks to extrapolate periodic functions.
problem Extrapolating periodic functions without prior knowledge of their form.
method Normalized Base-2 Encoding (NB2E) for continuous numerical values.
result MLPs using NB2E can successfully extrapolate diverse periodic signals.
Fluctuation identities with continuous monitoring and their application to price barrier optionsq-fin.CP
We present a numerical scheme to calculate fluctuation identities for exponential Lévy processes in the continuous monitoring case. This includes the Spitzer identities for touching a single upper or lower barrier, and the more difficult case of the two-barriers exit problem. These identities are given in the Fourier-L…
Least squares estimation for the subcritical Heston model based on continuous time observationsmath.ST
We prove strong consistency and asymptotic normality of least squares estimators for the subcritical Heston model based on continuous time observations. We also present some numerical illustrations of our results.
Numeracy is the ability to understand and work with numbers. It is a necessary skill for composing and understanding documents in clinical, scientific, and other technical domains. In this paper, we explore different strategies for modelling numerals with language models, such as memorisation and digit-by-digit composi…
In this paper, we present a novel penalty approach for the numerical solution of continuously controlled HJB equations and HJB obstacle problems. Our results include estimates of the penalisation error for a class of penalty terms, and we show that variations of Newton's method can be used to obtain globally convergent…
Extends co-clustering to mixed numerical and binary data.
problem Co-clustering of mixed data types (numerical and binary).
method Latent block models for mixed data types.
result Effectiveness of the proposed approach on simulated data.
ContinuousNet generalizes ResNets to continuous dynamical systems.
problem ResNets fail to be meaningful dynamical integrators.
method Embedding continuous dynamical systems into higher-order numerical integration schemes (Runge Kutta).
result ContinuousNet exhibits invariance to discrete time step sizes and numerical integration schemes.
RANDPOL uses randomized networks for efficient reinforcement learning in continuous state and action MDPs.
problem Efficient reinforcement learning in environments with continuous state and action spaces.
method RANDPOL uses randomized function approximation to represent policy and value functions, providing finite time guarantees and improved numerical performance.
result RANDPOL achieves better numerical performance and provides finite time guarantees compared to deep neural network based algorithms.
New algorithm for aggregate inference in HMMs with continuous observations.
problem Inference in large populations with indistinguishable individuals and continuous measurements.
method Continuous observation collective forward-backward algorithm extending existing discrete case algorithm.
result Efficacy demonstrated through numerical experiments.
Discrete Gaussian noise preserves privacy and accuracy in differential privacy.
problem Finite computers cannot represent continuous Gaussian noise, leading to privacy breaches and loss of interpretability.
method Introduced and analyzed discrete Gaussian noise, providing privacy and accuracy guarantees similar to continuous Gaussian noise.
result Discrete Gaussian noise offers the same privacy and accuracy as continuous Gaussian noise, with efficient sampling algorithms.
In this paper, a finite-state mean-reverting model for the short-rate, based on the continuous time Ehrenfest process, will be examined. Two explicit pricing formulae for zero-coupon bonds will be derived in the general and the special symmetric cases. Its limiting relationship to the Vasicek model will be examined wit…
Study numerical invariants under retraction maps between topological spaces.
problem Understand behavior of invariants like cohomological dimensions under retractions.
method Introduced a notion of retraction and studied several numerical invariants.
result Proved inequalities between invariants hold under retractions.
QSurv models survival data without discretization, achieving high accuracy.
problem Intractable likelihood estimation for continuous-time survival models.
method QSurv uses numerical quadrature for cumulative hazard approximation and time-conditioned low-rank adaptation.
result QSurv achieves competitive predictive performance and interpretable hazard patterns.
Continuous time stochastic processes are useful models especially for financial and insurance purposes. The numerical simulation of such models is dependant of the time discrete discretization, of the parametric estimation and of the choice of a random number generator. The aim of this paper is to provide the tools for…
Generalizes PCA and ICA for continuous-time signals using neural networks.
problem Low-rank decomposition of continuous-time vector-valued signals.
method Implicit neural network framework to learn numerical approximations of PCA and ICA.
result Unified approach to PCA and ICA in continuous domain, enforcing decorrelation and independence.
Efficient method for learning continuous exponential families beyond Gaussian.
problem Learning continuous exponential families with unbounded support.
method Interaction Screening approach for scalable learning of continuous graphical models.
result Our estimator maintains similar accuracy and sample complexity scalings compared to alternative approaches, while improving run-time.
In this note we propose a new approach towards solving numerically optimal stopping problems via reinforced regression based Monte Carlo algorithms. The main idea of the method is to reinforce standard linear regression algorithms in each backward induction step by adding new basis functions based on previously estimat…
The theory of geometric zeta functions for locally symmetric spaces as initialized by Selberg and continued by numerous mathematicians is generalized to the case of higher rank spaces. We show analytic continuation, describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalize…
We consider a Hidden Markov Model (HMM) where the integrated continuous-time Markov chain can be observed at discrete time points perturbed by a Brownian motion. The aim is to derive a filter for the underlying continuous-time Markov chain. The recursion formula for the discrete-time filter is easy to derive, however i…
Study proposes Local Linear Encoding for better feature discretization.
problem Improving feature discretization for numeric data.
method Theoretical analysis and Local Linear Encoding (LLE) method.
result LLE outperforms conventional methods with fewer parameters.
Continuous formulation of machine learning models and algorithms.
problem Generalization error and implicit regularization in machine learning.
method Continuous formulation in calculus of variations and differential-integral equations, with new models and algorithms.
result Conventional models and algorithms can be recovered as particular discretizations.
This paper deals with the problem of large-scale linear supervised learning in settings where a large number of continuous features are available. We propose to combine the well-known trick of one-hot encoding of continuous features with a new penalization called \emph{binarsity}. In each group of binary features comin…
New method uses tensor trains for efficient PDE approximation.
problem High-dimensional PDEs and the curse of dimensionality.
method Tensor trains and backward stochastic differential equations for parabolic PDEs.
result Achieves a favorable trade-off between accuracy and computational efficiency.
LFlows model fluid densities and velocities using invertible maps that satisfy the continuity equation.
problem Modeling fluid densities and velocities continuously in space and time.
method LFlows are based on invertible maps that satisfy the continuity equation, derived from classical theory of Lagrangian flows for smooth vector fields.
result LFlows show higher predictive accuracy in density modeling tasks compared to competing models in 2D and 3D.
Paper proposes copula-based models for analyzing multivariate zero-inflated continuous data.
problem Challenges in analyzing multivariate zero-inflated continuous data with mixed discreteness and continuity.
method Proposes two copula-based density estimation models and rectified Gaussian copula.
result Demonstrates superior performance compared to conventional methods.
In mathematical Finance calculating the Greeks by Malliavin weights has proved to be a numerically satisfactory procedure for finite-dimensional Itô-diffusions. The existence of Malliavin weights relies on absolute continuity of laws of the projected diffusion process and a sufficiently regular density. In this article…
A New Type Of Upper And Lower Bounds On Right-Tail Probabilities Of Continuous Random Variablesmath.PR
New bounds on continuous random variables' right-tail probabilities.
problem Finding precise upper and lower limits for right-tail probabilities of continuous random variables.
method Developed new bounds based on PDF, first derivative, and two parameters.
result The new bounds are tight for various continuous random variables.
Hypersolvers enable fast continuous-depth models for practical applications.
problem Infinite-depth models like Neural ODEs are computationally infeasible for large problems.
method Introducing hypersolvers, neural networks that solve ODEs efficiently with theoretical guarantees.
result Hypersolvers achieve comparable inference time to traditional discrete networks, making continuous-depth models practical.
New method optimizes SDE models using continuous-time gradient descent.
problem Optimizing over the stationary distribution of SDE models.
method Continuous-time stochastic gradient descent for SDE models.
result Asymptotic convergence to the direction of steepest descent.
The thesis examines stochastic calculus in option pricing with logistic models and numerical methods.
problem Exploring the application of stochastic calculus in option pricing.
method Monte-Carlo Simulation and machine learning algorithms.
result Insights from Peter Carr and Lorenzo Torricelli's convex duality in continuous models.
Analytical pricing formulas and Greeks are obtained for European and American basket put options using Mellin transforms. We assume assets are driven by geometric Brownian motion which exhibit correlation and pay a continuous dividend rate. A novel approach to numerical Mellin inversion is achieved via the fast Fourier…
Avanzi et al. (2016) recently studied an optimal dividend problem where dividends are paid both periodically and continuously with different transaction costs. In the Brownian model with Poissonian periodic dividend payment opportunities, they showed that the optimal strategy is either of the pure-continuous, pure-peri…
We consider an investor facing a classical portfolio problem of optimal investment in a log-Brownian stock and a fixed-interest bond, but constrained to choose portfolio and consumption strategies that reduce a dynamic shortfall risk measure. For continuous- and discrete-time financial markets we investigate the loss i…
Researchers study the normalizing constant of a continuous categorical distribution.
problem Understanding the normalizing constant of the continuous categorical distribution.
method Characterize numerical behavior and present theoretical and methodological advances.
result The normalizing constant can be written in closed form using elementary functions.
As energy markets begin clearing at sub-hourly rates, their interaction with load control systems becomes a potentially important consideration. A simple model for the control of thermal systems using market-based power distribution strategies is proposed, with particular attention to the behavior and dynamics of elect…
Cai, Song and Kou (2015) [Cai, N., Y. Song, S. Kou (2015) A general framework for pricing Asian options under Markov processes. Oper. Res. 63(3): 540-554] made a breakthrough by proposing a general framework for pricing both discretely and continuously monitored Asian options under one-dimensional Markov processes. In …
CGNNs use wavelets for continuous function generation in infinite-dimensional spaces.
problem Generating continuous functions in infinite-dimensional spaces for applications like inverse problems.
method Inspired by DCGAN, CGNNs use wavelet multiresolution analysis with convolutional and nonlinear layers.
result CGNNs can be injective under certain conditions on filters and nonlinearity, leading to Lipschitz stability estimates.
We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constan…
Develops DPG methods for continuous-time RL with deterministic policies.
problem High variance and slow convergence in stochastic policy RL methods.
method Derives continuous-time policy gradient formula and proposes CT-DDPG algorithm.
result CT-DDPG achieves superior stability and faster convergence in continuous-time RL.
We consider the problem of numerical approximation for forward-backward stochastic differential equations with drivers of quadratic growth (qgFBSDE). To illustrate the significance of qgFBSDE, we discuss a problem of cross hedging of an insurance related financial derivative using correlated assets. For the convergence…
The paper defines cross-section continuity for angular momentum definitions and finds the CWY definition valid.
problem Defining angular momentum at null infinity and ensuring its continuity across different cross-sections.
method Introducing cross-section continuity as a criterion and proving it for specific angular momentum definitions.
result The Chen-Wang-Yau definition of angular momentum satisfies cross-section continuity, while the Compere-Nichols modification does not.
Theoretical and Numerical Analysis of an Optimal Execution Problem with Uncertain Market Impactq-fin.TR
This paper is a continuation of Ishitani and Kato (2015), in which we derived a continuous-time value function corresponding to an optimal execution problem with uncertain market impact as the limit of a discrete-time value function. Here, we investigate some properties of the derived value function. In particular, we …
Paper optimizes portfolios for absolute return funds with constraints.
problem Optimizing portfolios with constraints for absolute return funds.
method Stochastic control framework with numerical solution using kernel-based collocation method.
result Leverage is necessary to achieve the target level.