arXiv subjects

Torkel E Loman

Publications and source records attributed to Torkel E Loman.

3 recordsLinked to original sources

arXiv2026

Structural functional identifiability and model discovery in differential equation models

Differential equation models are widely used to describe, interpret, and predict dynamical phenomena across science and engineering. In practice, however, the governing dynamics are rarely fully known and must be inferred from observational data. Traditionally, inverse problems in differential equation modelling have focused on estimating unknown parameter values. In this setting, structural identifiability determines whether parameter values can, in principle, be uniquely recovered from ideal observations and is, therefore, a prerequisite for meaningful inference. More recently, the integration of machine learning with mechanistic modelling has enabled the discovery of unknown equations, functions, and constitutive relationships, substantially expanding the space of admissible models. This raises a fundamental question: under what conditions can unknown functional components be uniquely recovered from data? In this paper, we generalise the classical notion of structural parameter identifiability to functional identifiability. We first identify broad classes of models for which unique functional recovery is impossible. We then show how functional identifiability can be assessed for differential equation models using differential algebra-based techniques which are well-established as a means of assessing structural identifiability for ordinary differential equation-based models. Our framework reveals new phenomena that arise in the transition from parametric to functional inference and have no analogue in the classical setting. Finally, we characterise functional identifiability in several common model classes. Taken together, our results demonstrate that functional identifiability provides a theoretical foundation for modern inverse problems in differential equation modelling, particularly those that use machine learning representations of unknown system components.

math.ST↗

arXiv2025

Exact identifiability analysis for a class of partially observed near-linear stochastic differential equation models

Stochasticity plays a key role in many biological systems, necessitating the calibration of stochastic mathematical models to interpret associated data. For model parameters to be estimated reliably, it is typically the case that they must be structurally identifiable. Yet, while theory underlying structural identifiability analysis for deterministic differential equation models is highly developed, there are currently no tools for the general assessment of stochastic models. In this work, we present a differential algebra-based framework for the structural identifiability analysis of linear and a class of near-linear partially observed stochastic differential equation (SDE) models. Our framework is based on a deterministic recurrence relation that describes the dynamics of the statistical moments of the system of SDEs. From this relation, we iteratively form a series of necessarily satisfied equations involving only the observed moments, from which we are able to establish structurally identifiable parameter combinations. We demonstrate our framework for a suite of linear (two- and $n$-dimensional) and non-linear (two-dimensional) models. Most importantly, we define the notion of structural identifiability for SDE models and establish the effect of the initial condition on identifiability. We conclude with a discussion on the applicability and limitations of our approach, and potential future research directions in this understudied area.

stat.ME↗

arXiv2025

Functional and parametric identifiability for universal differential equations applied to chemical reaction networks

Mathematical modelling has traditionally relied on detailed system knowledge to construct mechanistic models. However, the advent of large-scale data collection and advances in machine learning have led to an increasing use of data-driven approaches. Recently, hybrid models have emerged that combine both paradigms: well-understood system components are modelled mechanistically, while unknown parts are inferred from data. Here, we focus on one such class: universal differential equations (UDEs), where neural networks are embedded within differential equations to approximate unknown dynamics. When fitted to data, these networks act as universal function approximators, learning missing functional components. In this work, we note that UDE identifiability, i.e. our ability to identify true system properties, can be split into parametric and functional identifiability (assessing identifiability for the mechanistic and data-driven model parts, respectively). Next, we investigate how UDE properties, such as neural network numbers and constraints, affect parametric and functional identifiability. Notably, we show that across a wide range of models, the generalisation of a fully mechanistic model to a UDE has little impact on the mechanistic components' parametric identifiability. Finally, we note that hybrid modelling through the fitting of unknown functions (as achieved by UDEs) is particularly well-suited to chemical reaction network (CRN) modelling. Here, CRNs are used in fields ranging from systems biology, chemistry, and pharmacology to epidemiology and population dynamics, making them highly relevant for study. By showcasing how CRN-based UDE models can be highly interpretable, we demonstrate that this hybrid approach is a promising avenue for future applications.

math.DS↗