Dimensional Reduction of Dynamical Systems by Machine Learning - Automatic Generation of a Macroscopic Model

Aug 5, 2026

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A framework is introduced to generate phenomenological macroscopic models from microscopic dynamical systems with large degrees of freedom using machine learning. The method jointly searches for an interpretable projection to extensive macroscopic variables and the time-evolution equation governing them by minimizing a residual error between predicted and observed next-step macrostates. Indeterminacy in the learned projection is handled by structural restrictions and normalization constraints. Demonstrations are provided using elementary cellular automata and their spatiotemporal dynamics.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages9

Reading Time23 min

AccessFree to download

Abstract

Problem setting: microscopic to macroscopic dynamical systems

Learning objective and residual error

Regression when projection is given

Joint learning when projection is unknown

Example: elementary cellular automata

System definition and update rule

Indeterminacy of projection and constraints

Parametric forms and function approximation

Block-based extensive variables

Polynomial form of the macroscopic evolution

arXiv :2006 .04482v2 [ cond-mat .stat-mech] 15 Jun 2020 Dimensional Reduction of Dynamical Systems by Machine Learning: Automatic Generation of a Macroscopic Model Tomoaki Nogawa􀀃 Faculty of Medicine, Toho University, 5-21-16, Omori-Nishi, Ota-ku, Tokyo, 143-8540, Japan (Dated: June 16, 2020) Abstract We propose the framework to generate a phenomenological model that extract the essence of a dynamical system with large degrees of freedom by using machine learning. For a given microscopic dynamical system, we simultaneously seek for the suitable projection to a macroscopic variable, which is supposed to be extensive, and the time proceeding equation that governs them. The utility of this method is demonstrated by the application to the elementary cellular automata. In the study of the dynamics of a complex system with large degrees of freedom, we often use a phenomenological model with small degrees of freedom, e.g., the Lorentz model for atmospheric variability [1] . Once such a model is supposed, we have various analysis techniques developed in the 􀀌eld of so-called nonlinear dynamics. Usually, such a reduced model is empirically introduced with a drastic approximation owing to the intuition and insight of researchers. Although it is an orthodox task for statistical physicists to derive such a model ab initio from a basic microscopic model, it is impossible in most cases. The aim of this paper is to propose a generic framework to generate a macroscopic dynamical system (DS) for a given microscopic DS with help from machine learning. One of the key feature of the modern machine learning, represented by deep neural networks [2], is automatic extraction of feature amount of the object. In the similar manner, we try to 􀀌nd the suitable variable to describe the dynamics of a many-body system. The DS that the obtained variables obey is also a product of the learning. Let us start with a microscopic DS xt+1 = f(xt) 2 RN ; N 􀀝 1 : (1) The goal is obtaining a macroscopic DS in the form X (xt+1) = F (X(xt )) 2 Rn ; n 􀀜 N; (2) where X : RN ! Rn is a projection to macroscopic variable and F : Rn ! Rn is time proceeding operation. It is desired that X is interpretable for human being. For this goal, we use machine learning in which a data point is (xt ; xt+1) 2 RN 􀀂 2 satisfying Eq. (1) . More concretely, we seek for F and X that minimize the error functional E [F ; X] := j F (X(xt )) 􀀀 X(xt+1) j 2 =n: (3) Hereafter, the overline denotes the average over data points. If E [F ; X] equals zero, the obtained DS is exactly closed. Otherwise, it is an approximated formula, whose precision is evaluated by the residual error. The error generally decreases with n. This n-dependence would be a clue to quantify the complexity or degree of nonequilibrium of the system. If X is given, the task is simple regression of F in the from of supervised learning. There are a lot of studies on this kind of regression of DS [3, 4] . If X is not given, it is not a popular problem. This is, however, similar to the 􀀌nite size scaling [5, 6], where we seek for both how to scale the variables and what equation the scaled variables satisfy. In the following, we employ the elementary cellular automata (ECA) [7] as an example. The system has binary variables: xit 2 f0 ; 1g where i 2 f1 ; 􀀁 􀀁 􀀁 ; Ng and t 2 f0 ; 1 ; 􀀁 􀀁 􀀁 g denote the position in one-dimensional chain and time, respectively. Time evolution is driven by the rule in which xi;t+1 is determined by three variables: (xi􀀀1;t , xit , xi+1;t) . We impose the periodic boundary condition for i. The number of the possible rules is 223 = 256 . There exists inde􀀌niteness in X; If (X ; F) satis􀀌es Eq. (2), (G 􀀎 X ; G 􀀎 F 􀀎 G􀀀1) with arbitrary G : Rn ! Rn that has an inverse function does too. To avoid this inde􀀌niteness, we need to impose some restrictions on X. Here, we suppose that each component of X takes the form 1 Nb Xm (x) = Nb 􀀘m (yj) ; yj 2 Rb ; yj` = xb (j􀀀1)+`: (4)

FAQs

What does the proposed framework aim to achieve?

It generates a macroscopic dynamical system that captures the essence of a microscopic system with many degrees of freedom, using machine learning.

How are the projection X and macroscopic evolution F learned?

They are learned simultaneously by minimizing an error functional between F applied to X(xt) and the observed X(xt+1).

Why is indeterminacy of the learned projection a concern, and how is it addressed?

Different transformations can preserve the macroscopic equation, so restrictions are imposed on X via block-structured extensive variables and normalization constraints.

Aug 5, 2026

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