Related papers: A Taxonomy of Numerical Differentiation Methods
We introduce the concept of numerical Gaussian processes, which we define as Gaussian processes with covariance functions resulting from temporal discretization of time-dependent partial differential equations. Numerical Gaussian processes,…
Data segmentation a.k.a. multiple change point analysis has received considerable attention due to its importance in time series analysis and signal processing, with applications in a variety of fields including natural and social sciences,…
Derivatives, mostly in the form of gradients and Hessians, are ubiquitous in machine learning. Automatic differentiation (AD), also called algorithmic differentiation or simply "autodiff", is a family of techniques similar to but more…
Differential equations are a ubiquitous tool to study dynamics, ranging from physical systems to complex systems, where a large number of agents interact through a graph with non-trivial topological features. Data-driven approximations of…
Mathematical models allow us to gain a deeper understanding of real-world dynamical systems. One of the most powerful mathematical frameworks for modeling real-world phenomena are systems of differential equations. In the majority of fields…
Differential privacy has been an exceptionally successful concept when it comes to providing provable security guarantees for classical computations. More recently, the concept was generalized to quantum computations. While classical…
The dynamics of physical theories is usually described by differential equations. Difference equations then appear mainly as an approximation which can be used for a numerical analysis. As such, they have to fulfill certain conditions to…
A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full…
Pattern discovery algorithms in the music domain aim to find meaningful components in musical compositions. Over the years, although many algorithms have been developed for pattern discovery in music data, it remains a challenging task. To…
Differentially private (DP) mechanisms face the challenge of providing accurate results while protecting their inputs: the privacy-utility trade-off. A simple but powerful technique for DP adds noise to sensitivity-bounded query outputs to…
An efficient coordinate-free notation is elucidated for differentiating matrix expressions and other functions between higher-dimensional vector spaces. This method of differentiation is known, but not explained well, in the literature.…
We propose robust methods to identify underlying Partial Differential Equation (PDE) from a given set of noisy time dependent data. We assume that the governing equation is a linear combination of a few linear and nonlinear differential…
Many problems in Physics and Chemistry are formulated as the minimization of a functional. Therefore, methods for solving these problems typically require differentiating maps whose input and/or output are functions -- commonly referred to…
The performance of machine learning models relies heavily on the quality of input data, yet real-world applications often face significant data-related challenges. A common issue arises when curating training data or deploying models: two…
Time series models are ubiquitous in science, arising in any situation where researchers seek to understand how a system's behaviour changes over time. A key problem in time series modelling is \emph{inference}; determining properties of…
Writing correct distributed programs is hard. In spite of extensive testing and debugging, software faults persist even in commercial grade software. Many distributed systems, especially those employed in safety-critical environments,…
Discovering nonlinear differential equations that describe system dynamics from empirical data is a fundamental challenge in contemporary science. Here, we propose a methodology to identify dynamical laws by integrating denoising techniques…
The numerical methods for differential equation solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods have the restricted class of…
The statistical physics approach to the number partioning problem, a classical NP-hard problem, is both simple and rewarding. Very basic notions and methods from statistical mechanics are enough to obtain analytical results for the phase…
We have presented some practical consequences on the molecular-dynamics simulations arising from the numerical algorithm published recently in paper Int. J. Mod. Phys. C 16, 413 (2005). The algorithm is not a finite-difference method and…