Bayesian uncertainty quantification for data-driven equation learning

TL;DR

This paper investigates how observation noise affects the uncertainty in data-driven differential equation models and proposes a Bayesian approach to quantify this uncertainty while learning the correct models from noisy data.

Contribution

It introduces a method combining equation learning with approximate Bayesian computation to quantify uncertainty in models learned from noisy data.

Findings

Successful learning of differential equations from noisy data.

Uncertainty in model structure and parameters can be quantified.

Bayesian posterior distributions provide insights into model confidence.

Abstract

Equation learning aims to infer differential equation models from data. While a number of studies have shown that differential equation models can be successfully identified when the data are sufficiently detailed and corrupted with relatively small amounts of noise, the relationship between observation noise and uncertainty in the learned differential equation models remains unexplored. We demonstrate that for noisy data sets there exists great variation in both the structure of the learned differential equation models as well as the parameter values. We explore how to combine data sets to quantify uncertainty in the learned models, and at the same time draw mechanistic conclusions about the target differential equations. We generate noisy data using a stochastic agent-based model and combine equation learning methods with approximate Bayesian computation (ABC) to show that the correct differential equation model can be successfully learned from data, while a quantification of uncertainty is given by a posterior distribution in parameter space.