Related papers: Combining fixpoint and differentiation theory
Fixpoint operators are tools to reason on recursive programs and data types obtained by induction (e.g. lists, trees) or coinduction (e.g. streams). They were given a categorical treatment with the notion of categories with fixpoints. A…
Incremental computation has recently been studied using the concepts of change structures and derivatives of programs, where the derivative of a function allows updating the output of the function based on a change to its input. We…
Approximation Fixpoint Theory (AFT) is an algebraic framework designed to study the semantics of non-monotonic logics. Despite its success, AFT is not readily applicable to higher-order definitions. To solve such an issue, we devise a…
Cartesian differential categories are categories equipped with a differential combinator which axiomatizes the directional derivative. Important models of Cartesian differential categories include classical differential calculus of smooth…
This article proposes new perspectives for developing derivative based numerical algorithms, supported by the introduction of a generalized derivative operators. It demonstrates that these operators have the potential to enhance and extend…
Cartesian differential categories are categories equipped with a differential combinator which axiomatizes the directional derivative. Important models of Cartesian differential categories include classical differential calculus of smooth…
The unprecedented pace of machine learning research has lead to incredible advances, but also poses hard challenges. At present, the field lacks strong theoretical underpinnings, and many important achievements stem from ad hoc design…
The goal of this thesis is threefold: first, to provide a general semantic setting for reasoning about incremental computation. Second, to establish and clarify the connection between derivatives in the incremental sense and derivatives in…
This paper provides an overview of Lawvere's Fixed-Point Theorem in category theory and aims to detail the universal framework underlying self-reference and recursive structures. First, we rigorously define fundamental concepts - such as…
We develop some aspects of the theory of derivators, pointed derivators, and stable derivators. As a main result, we show that the values of a stable derivator can be canonically endowed with the structure of a triangulated category.…
Motivated by an ongoing project on computer aided derivation of asymptotic models governed by partial differential equations, we introduce a class of term transformations that consists of traversal strategies and insertion of contexts. We…
Cartesian differential categories come equipped with a differential combinator that formalizes the directional derivative from multivariable calculus. Cartesian differential categories provide a categorical semantics of the differential…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
In 2017, Bauer, Johnson, Osborne, Riehl, and Tebbe (BJORT) showed that the Abelian functor calculus provides an example of a Cartesian differential category. The definition of a Cartesian differential category is based on a differential…
Considering the large number of fractional operators that exist, and since it does not seem that their number will stop increasing soon at the time of writing this paper, it is presented for the first time, as far as the authors know, a…
This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e. the original problem is first reformulated as a nonconvex optimization problem, its well-posedness…
Category theory has been successfully applied in various domains of science, shedding light on universal principles unifying diverse phenomena and thereby enabling knowledge transfer between them. Applications to machine learning have been…
Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…
Recent theoretical work on automatic differentiation (autodiff) has focused on characteristics such as correctness and efficiency while assuming that all derivatives are automatically generated by autodiff using program transformation, with…
We give a quick survey of the various fixed point theorems in computability theory, partial combinatory algebra, and the theory of numberings, as well as generalizations based on those. We also point out several open problems connected to…