Extracting dynamical equations from experimental data is NP-hard

TL;DR

This paper proves that extracting the underlying dynamical equations from experimental data is computationally NP-hard for both classical and quantum systems, highlighting fundamental limits in data-driven discovery of physical laws.

Contribution

It establishes the NP-hardness of identifying dynamical equations from data, providing complexity-theoretic insights into classical and quantum embedding problems.

Findings

Extraction of dynamical equations is NP-hard.

Provides complexity-theoretic solutions to classical and quantum embedding problems.

Highlights computational limits in discovering physical laws from data.

Abstract

The behavior of any physical system is governed by its underlying dynamical equations. Much of physics is concerned with discovering these dynamical equations and understanding their consequences. In this work, we show that, remarkably, identifying the underlying dynamical equation from any amount of experimental data, however precise, is a provably computationally hard problem (it is NP-hard), both for classical and quantum mechanical systems. As a by-product of this work, we give complexity-theoretic answers to both the quantum and classical embedding problems, two long-standing open problems in mathematics (the classical problem, in particular, dating back over 70 years).