Differentiation methods as a systematic uncertainty source in equation discovery
Maria Khilchuk, Ilya Markov, Alexander Hvatov
TL;DR
This work reveals that numerical differentiation, a routine prestep in differential equation discovery, is a major source of systematic uncertainty that can bias both equation form and parameter estimates. By evaluating six differentiation techniques across several ODE/PDE systems with SINDy and EPDE, the study shows that higher derivative precision can amplify noise and create spurious features, while smoothing approaches often yield more robust structural recovery. The key contribution is treating differentiation as a core modeling choice and advocating ensemble-based discovery that diversifies differentiation methods to mitigate method-induced biases. The findings have practical implications for improving reliability of data-driven DE discovery in noisy, real-world datasets and motivate integrating differentiation heterogeneity into discovery pipelines.
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.