相关论文: Jet Functors and Weil Algebras in Automatic Differ…
We propose an algorithm to compute the $C^\infty$-ring structure of arbitrary Weil algebra. It allows us to do some analysis with higher infinitesimals numerically and symbolically. To that end, we first give a brief description of the…
Automatic differentiation (AD) has been a topic of interest for researchers in many disciplines, with increased popularity since its application to machine learning and neural networks. Although many researchers appreciate and know how to…
This article provides an overview of some of the mathematical principles of Automatic Differentiation (AD). In particular, we summarise different descriptions of the Forward Mode of AD, like the matrix-vector product based approach, the…
We present semantic correctness proofs of automatic differentiation (AD). We consider a forward-mode AD method on a higher-order language with algebraic data types and we characterise it as the unique structure-preserving macro given a…
Automatic differentiation (AD) is conventionally understood as a family of distinct algorithms, rooted in two "modes" -- forward and reverse -- which are typically presented (and implemented) separately. Can there be only one? Following up…
Automatic differentiation (AD) is a technique for computing the derivative of a function represented by a program. This technique is considered as the de-facto standard for computing the differentiation in many machine learning and…
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…
Automatic differentiation (AD) in reverse mode (RAD) is a central component of deep learning and other uses of large-scale optimization. Commonly used RAD algorithms such as backpropagation, however, are complex and stateful, hindering deep…
In scientific computation, it is often necessary to calculate higher-order derivatives of a function. Currently, two primary methods for higher-order automatic differentiation exist: symbolic differentiation and algorithmic automatic…
Automatic differentiation (AD) is an ensemble of techniques that allow to evaluate accurate numerical derivatives of a mathematical function expressed in a computer programming language. In this paper we use AD for stating and solving solid…
We construct the Weil functor $T^A$ corresponding to a general Weil algebra $A = K \oplus N$: this is a functor from the category of manifolds over a general topological base field or ring $K$ (of arbitrary characteristic) to the category…
We prove that there exists a version of Weil descent, or Weil restriction, in the category of $\mathcal{D}$-algebras. The objects of this category are $k$-algebras $R$ equipped with a homomorphism $e \colon R \to R \otimes_k \mathcal{D}$…
In this paper we give an axiomatization of differential geometry comparable to model categories for homotopy theory. Weil functors play a predominant role.
We present semantic correctness proofs of Automatic Differentiation (AD). We consider a forward-mode AD method on a higher order language with algebraic data types, and we characterise it as the unique structure preserving macro given a…
Automatic Differentiation (AD) is a powerful tool that allows calculating derivatives of implemented algorithms with respect to all of their parameters up to machine precision, without the need to explicitly add any additional functions.…
Machine learning and neural network models in particular have been improving the state of the art performance on many artificial intelligence related tasks. Neural network models are typically implemented using frameworks that perform…
We introduce Combinatory Homomorphic Automatic Differentiation (CHAD), a principled, pure, provably correct define-then-run method for performing forward- and reverse-mode automatic differentiation (AD) on programming languages with…
In mathematics and computer algebra, automatic differentiation (AD) is a set of techniques to evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how…
The method of direct calculation of the group of $\mathbb R$-algebra automorphisms of a Weil algebra is presented in detail. The paper is focused on the case of a one-componental group and presents two cases of values of the determinant of…
We present ForwardDiff, a Julia package for forward-mode automatic differentiation (AD) featuring performance competitive with low-level languages like C++. Unlike recently developed AD tools in other popular high-level languages such as…