A Bayesian Approach for Discovering Time- Delayed Differential Equation from Data

TL;DR

This paper introduces BayTiDe, a Bayesian method that accurately discovers time-delayed differential equations from data, even with large delays and noisy measurements, improving robustness and computational efficiency over existing techniques.

Contribution

The paper presents a novel Bayesian framework with a sparsity prior for identifying large time delays in differential equations from data, enhancing scalability and robustness.

Findings

Successfully recovers delayed differential equations from noisy data.

Accurately identifies large time delays proportional to data resolution.

Demonstrates computational efficiency and robustness across numerical examples.

Abstract

Time-delayed differential equations (TDDEs) are widely used to model complex dynamic systems where future states depend on past states with a delay. However, inferring the underlying TDDEs from observed data remains a challenging problem due to the inherent nonlinearity, uncertainty, and noise in real-world systems. Conventional equation discovery methods often exhibit limitations when dealing with large time delays, relying on deterministic techniques or optimization-based approaches that may struggle with scalability and robustness. In this paper, we present BayTiDe - Bayesian Approach for Discovering Time-Delayed Differential Equations from Data, that is capable of identifying arbitrarily large values of time delay to an accuracy that is directly proportional to the resolution of the data input to it. BayTiDe leverages Bayesian inference combined with a sparsity-promoting discontinuous spike-and-slab prior to accurately identify time-delayed differential equations. The approach accommodates arbitrarily large time delays with accuracy proportional to the input data resolution, while efficiently narrowing the search space to achieve significant computational savings. We demonstrate the efficiency and robustness of BayTiDe through a range of numerical examples, validating its ability to recover delayed differential equations from noisy data.