Multi-Task Equation Discovery
S C Bee, N Dervilis, K Worden, L A Bull
TL;DR
This work tackles the challenge of generalising equation-discovery models across different operating conditions by using a Bayesian multi-task relevance vector machine (RVM) to simultaneously identify governing terms from multiple excitation datasets. By sharing a weight representation across related tasks while allowing task-specific noise, the method improves the recovery of nonlinear dynamics when individual datasets are insufficiently excited. In a simulated SDOF oscillator with linear and cubic stiffness, the multi-task RVM outperforms single-task approaches, particularly at medium excitation, and demonstrates lower variance in predictive accuracy, highlighting enhanced generalisation for structural health monitoring. The study shows that leveraging cross-task information can mitigate over-fitting and yield more robust, physics-faithful models applicable to SHM under varying loads.
Abstract
Equation discovery provides a grey-box approach to system identification by uncovering governing dynamics directly from observed data. However, a persistent challenge lies in ensuring that identified models generalise across operating conditions rather than over-fitting to specific datasets. This work investigates this issue by applying a Bayesian relevance vector machine (RVM) within a multi-task learning (MTL) framework for simultaneous parameter identification across multiple datasets. In this formulation, responses from the same structure under different excitation levels are treated as related tasks that share model parameters but retain task-specific noise characteristics. A simulated single degree-of-freedom oscillator with linear and cubic stiffness provided the case study, with datasets generated under three excitation regimes. Standard single-task RVM models were able to reproduce system responses but often failed to recover the true governing terms when excitations insufficiently stimulated non-linear dynamics. By contrast, the MTL-RVM combined information across tasks, improving parameter recovery for weakly and moderately excited datasets, while maintaining strong performance under high excitation. These findings demonstrate that multi-task Bayesian inference can mitigate over-fitting and promote generalisation in equation discovery. The approach is particularly relevant to structural health monitoring, where varying load conditions reveal complementary aspects of system physics.