相关论文: TaylUR, an arbitrary-order diagonal automatic diff…

The problem of inverting the total divergence operator is central to finding components of a given conservation law. This might not be taxing for a low-order conservation law of a scalar partial differential equation, but integrable systems…

This paper is concerned with the efficient evaluation of higher-order derivatives of functions $f$ that are composed of matrix operations. I.e., we want to compute the $D$-th derivative tensor $\nabla^D f(X) \in \mathbb R^{N^D}$, where…

Forward Automatic Differentiation (AD) is a technique for augmenting programs to compute derivatives. The essence of Forward AD is to attach perturbations to each number, and propagate these through the computation. When derivatives are…

Differential machine learning (DML) is a recently proposed technique that uses samplewise state derivatives to regularize least square fits to learn conditional expectations of functionals of stochastic processes as functions of state…

We propose a differentiable nonparametric algorithm, the Delaunay triangulation learner (DTL), to solve the functional approximation problem on the basis of a $p$-dimensional feature space. By conducting the Delaunay triangulation algorithm…

Computers calculate transcendental functions by approximating them through the composition of a few limited-precision instructions. For example, an exponential can be calculated with a Taylor series. These approximation methods were…

Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly…

In this paper we describe an algorithm and a Fortran 90 module ({\bf Conical}) for the computation of the conical function $P^m_{-\tfrac12+i\tau}(x)$ for $x>-1$, $m \ge 0$, $\tau >0$. These functions appear in the solution of Dirichlet…

While the definition of a fractional integral may be codified by Riemann and Liouville, an agreed-upon fractional derivative has eluded discovery for many years. This is likely a result of integral definitions including numerous constants…

The first-order Euler-Maclaurin formula relates the sum of the values of a smooth function on an interval of integers with its integral on the same interval on $\mathbb R$. We formulate here the analogue for functions that are just of…

This paper introduces a new computational framework to derive electromagnetic field derivatives with respect to multiple design parameters up to any order with the Finite-Difference Time-Domain (FDTD) technique. Specifically, only one FDTD…

Recently there has been a renewed interest in asymptotic Euler-MacLaurin formulas, partly due to applications to spectral theory of differential operators. Using elementary means, we recover such formulas for compactly supported smooth…

It is well-known that the two-parameter Mittag-Leffler (ML) function plays a key role in Fractional Calculus. In this paper, we address the problem of computing this function, when its argument is a square matrix. Effective methods for…

Classical functional calculus is primarily spectral, capturing eigenvalue information through resolvent methods while largely ignoring nilpotent structure. Building on the projector-nilpotent characterization developed in our companion…

We extend JAX with the capability to automatically differentiate higher-order functions (functionals and operators). By representing functions as a generalization of arrays, we seamlessly use JAX's existing primitive system to implement…

Differential equations parameterized by neural networks become expensive to solve numerically as training progresses. We propose a remedy that encourages learned dynamics to be easier to solve. Specifically, we introduce a differentiable…

We present a new Fortran library to evaluate all harmonic polylogarithms up to weight four numerically for any complex argument. The algorithm is based on a reduction of harmonic polylogarithms up to weight four to a minimal set of basis…

Holderian functions have strong non-linearities, which result in singularities in the derivatives. This manuscript presents several fractional-order Taylor expansions of H\"olderian functions around points of non- differentiability. These…

Almost all problems in applied mathematics, including the analysis of dynamical systems, deal with spaces of real-valued functions on Euclidean domains in their formulation and solution. In this paper, we describe the the tool Ariadne,…

The computation of higher order processes very often involves a large number of diagrams. In addition, it is in general not possible to solve the occurring integrals explicitly and expansions in small quantities have to be performed. This…