Related papers: Tropical differential equations
We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs), with a high degree of accuracy and up to a desired tolerance. We develop a differentiable…
Finding a common factor of two multivariate polynomials with approximate coefficients is a problem in symbolic-numeric computing. Taking a tropical view on this problem leads to efficient preprocessing techniques, applying polyhedral…
In this paper we investigate the computational complexity of solving ordinary differential equations (ODEs) $y^{\prime}=p(y)$ over \emph{unbounded time domains}, where $p$ is a vector of polynomials. Contrarily to the bounded (compact) time…
This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do…
In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus…
The shellability of the boundary complex of an unbounded polyhedron is investigated. To this end, it is necessary to pass to a suitable compactification, e.g., by one point. This observation can be exploited to prove that any tropical…
This article surveys the burgeoning area at the intersection of dynamical systems theory and algorithms for NP-hard problems. Traditionally, computational complexity and the analysis of non-deterministic polynomial-time (NP)-hard problems…
We consider multidimensional optimization problems that are formulated in the framework of tropical mathematics to minimize functions defined on vectors over a tropical semifield (a semiring with idempotent addition and invertible…
A typical system of k difference (or differential) equations can be compressed, or folded into a difference (or ordinary differential) equation of order k. Such foldings appear in control theory as the canonical forms of the controllability…
In this note, we propose a novel technique to reduce the algorithmic complexity of neural network training by using matrices of tropical real numbers instead of matrices of real numbers. Since the tropical arithmetics replaces…
In recent years, there has been increasing interest in explanation methods for neural model predictions that offer precise formal guarantees. These include abductive (respectively, contrastive) methods, which aim to compute minimal subsets…
We determine a considerable class of nonlinear partial differential equation systems which have global regular solutions. Uniqueness is not a direct general consequence of this method. The scheme can be applied to the incompressible Navier…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
Some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored.
This paper depicts algorithms for solving the decision Boolean Satisfiability Problem. An extreme problem is formulated to analyze the complexity of algorithms and the complexity for solving it. A novel and easy reformulation as a lottery…
Classical approach of solvability problem has shed much light on what we can solve and what we cannot solve mathematically. Starting with quadratic equation, we know that we can solve it by the quadratic formula which uses square root.…
In this paper we develop a combinatorial abstraction of tropical linear programming. This generalizes the search for a feasible point of a system of min-plus-inequalities. It is based on the polyhedral properties of triangulations of the…
The method of this paper is my original creation. A new method for solving linear differential equations is proposed in this paper. The important conclusion of this paper is that arbitrary order linear ordinary differential equations with…
We study function-valued solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable parabolicity hypotheses. We provide…
The determinisation problem for min-plus (tropical) weighted automata was recently shown to be decidable. However, the proof is purely existential, relying on several non-constructive arguments. Our contribution in this work is twofold:…