Differentiation methods as a systematic uncertainty source in equation discovery

TL;DR

This paper investigates how the choice of differentiation methods in differential equation discovery introduces systematic uncertainty, affecting model accuracy and robustness, and emphasizes the need for careful method selection and ensemble approaches.

Contribution

It demonstrates that differentiation method variability significantly impacts equation discovery, highlighting the importance of method choice and proposing ensemble strategies to mitigate bias.

Findings

Different differentiation methods cause systematic biases in discovered equations.

High-resolution schemes can amplify measurement noise.

Regularized methods may obscure real physical variations.

Abstract

In differential equation discovery algorithms, numerical differentiation is usually a fixed preliminary step. Current methods improve robustness with data subsampling and sparsity but often ignore the variability from the differentiation method itself. We show that this choice systematically introduces uncertainty, affecting both equation form and parameter estimates. Our study indicates that high-resolution schemes can magnify measurement noise, while heavily regularized methods may mask real physical variations, which leads to method-dependent findings. By evaluating six differentiation techniques on various partial differential equations under diverse noise levels using SINDy and EPDE frameworks, we consistently notice methodological biases in the determined models. This underscores the importance of selecting differentiation methods as a key modeling choice and highlights a path to enhance ensemble-based discovery by diversifying methodologies.