Is a model equivalent to its computer implementation?

TL;DR

The paper questions whether a computer implementation is truly equivalent to its mathematical formulation, arguing that finite-precision arithmetic and execution choices introduce nontrivial differences, as illustrated by numbers like $1$ that cannot be represented exactly. It shows that implicit assumptions in code can propagate beyond the explicit model, challenging reproducibility even when code is published. The discussion extends to data-driven AI and quantum computing, where the model–implementation boundary shifts due to training dynamics, entanglement, and decoherence, complicating simulations on classical hardware. The authors advocate reimplementation of existing formulations, caution against implementation monoculture, and urge careful separation of mathematical content from software artifacts to preserve generalizable insights.

Abstract

A recent trend in mathematical modeling is to publish the computer code together with the research findings. Here we explore the formal question, whether and in which sense a computer implementation is distinct from the mathematical model. We argue that, despite the convenience of implemented models, a set of implicit assumptions is perpetuated with the implementation to the extent that even in widely used models the causal link between the (formal) mathematical model and the set of results is no longer certain. Moreover, code publication is often seen as an important contributor to reproducible research, we suggest that in some cases the opposite may be true. A new perspective on this topic stems from the accelerating trend that in some branches of research only implemented models are used, e.g., in artificial intelligence (AI). With the advent of quantum computers we argue that completely novel challenges arise in the distinction between models and implementations.

Figures (2)

• Figure 1: Schematic representation of the difference between (a) model implementations published along with the academic results and (b) the publication serving as the main source of information about the mathematical model. In (a) information flow and the flow of errors or the impact of design decisions, which are not part of the original mathematical model, are coupled, while in (b) due to the diversity of implementations by different modelers those decisions by a single modeler are not propagated.
• Figure 2: Summary of the concepts driving the distinction between models and their implementation. Left: Standard (historical) approach, where theory-driven mathematical models are explored on their own and with the help of computer simulations. This implements a cycle model $\rightarrow$ implementation $\rightarrow$ prediction $\rightarrow$ observations $\rightarrow$ model, which also challenges, and eventually contributes to, theory. The largest impact of this approach on modern computation is in the form of quantum computers. Right: New trend, where observations are directly translated into implementations (sometimes via a mediating mathematical model, which however stays in the background, because the implementation is regarded as the main outcome of the scientific endeavor). This process creates predictions, which can be compared to new observations. The largest impact of this approach on modern computation is in the form of artificial intelligence (AI).