Exact and Parametric Dynamical System Representation of Nonlinear Functions

2025/12/03 by Toshiyuki Ohtsuka, Ohtsuka, Toshiyuki
Computer Science · Engineering · #Algebraic number #Constant (computer programming) #Control Systems and Identification #Control and Stability of Dynamical Systems #Dynamical systems theory #Function (biology) #Nonlinear system #Parametric statistics #Polynomial and algebraic computation #Quadratic equation #Representation (politics)

paper · pdf · doi:10.48550/arxiv.2512.03779

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2025/12/03 · openalex created_date 2025/12/05 · openalex updated_date 2026/07/28

Abstract

Parametric representations of various functions are fundamental tools in science and engineering. This paper introduces a fixed-initial-state constant-input dynamical system (FISCIDS) representation, which provides an exact and parametric model for a broad class of nonlinear functions. A FISCIDS representation of a given nonlinear function consists of an input-affine dynamical system with a fixed initial state and constant input. The argument of the function is applied as the constant input to the input-affine system, and the value of the function is the output of the input-affine system at a fixed terminal time. We show that any differentially algebraic function has a quadratic FISCIDS representation. We also show that there exists an analytic function that is not differentially algebraic but has a quadratic FISCIDS representation. Therefore, most functions in practical problems in science and engineering can be represented by a quadratic FISCIDS representation.

Citations

• Exact and optimal quadratization of nonlinear finite-dimensional non-autonomous dynamical systems
• Augmented Neural ODEs
• FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models
• Glow: Generative Flow with Invertible 1x1 Convolutions
• Neural Ordinary Differential Equations
• Deep Neural Networks Motivated by Partial Differential Equations
• Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations
• Stable Architectures for Deep Neural Networks
• Variational Inference with Normalizing Flows
• Multilayer feedforward networks are universal approximators

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