Differential elimination for dynamical models via projections with applications to structural identifiability (2111.00991v3)

Abstract: Elimination of unknowns in a system of differential equations is often required when analysing (possibly nonlinear) dynamical systems models, where only a subset of variables are observable. One such analysis, identifiability, often relies on computing input-output relations via differential algebraic elimination. Determining identifiability, a natural prerequisite for meaningful parameter estimation, is often prohibitively expensive for medium to large systems due to the computationally expensive task of elimination. We propose an algorithm that computes a description of the set of differential-algebraic relations between the input and output variables of a dynamical system model. The resulting algorithm outperforms general-purpose software for differential elimination on a set of benchmark models from literature. We use the designed elimination algorithm to build a new randomized algorithm for assessing structural identifiability of a parameter in a parametric model. A parameter is said to be identifiable if its value can be uniquely determined from input-output data assuming the absence of noise and sufficiently exciting inputs. Our new algorithm allows the identification of models that could not be tackled before. Our implementation is publicly available as a Julia package at https://github.com/SciML/StructuralIdentifiability.jl.

• The paper introduces a novel projection-based elimination algorithm that reduces analysis time from hours to minutes compared to traditional methods.
• It proposes a randomized method for structural identifiability assessment that leverages projection-based representations to tackle previously intractable models.
• The approach applies advanced algebraic geometry to derive input-output relations, offering promising implications for parameter estimation and control design.

Differential Elimination for Dynamical Models via Projections with Applications to Structural Identifiability

This paper presents advancements in differential elimination methods for dynamical models, focusing on the development of projection-based techniques and their application to structural identifiability. The primary aim is to address the computational challenges associated with large systems of differential equations, particularly in deriving input-output relations essential for meaningful parameter estimation.

Main Contributions

The authors introduce a novel elimination algorithm that outperforms existing general-purpose software tailored for differential elimination. The algorithm's efficiency is showcased via a set of benchmark models, achieving significant computational gains—completed in minutes rather than hours compared to previous methodologies.

Alongside elimination, a randomized approach to assessing structural identifiability of dynamic models is proposed. This method enables tackling previously intractable models, extending the toolset available for structural identifiability analysis. Significantly, the novel approach utilizes projection-based representations of differential ideals, marking a departure from traditional syntactic representations like characteristic sets or Gröbner bases. This shift allows leveraging tools from constructive algebraic geometry, enhancing computational efficiency.

Theoretical Framework

The authors deploy an algebro-geometric perspective on differential equations, utilizing the notion of a projection-based representation. This involves examining the entire system of equations in an infinite-dimensional space, rationally parametrized by subsets of variables, then projecting this system to derive input-output relationships. The approach addresses the necessity for elimination techniques that are adaptive, strategically replacing state variables with output derivatives, ultimately providing a set of relationships pertinent to observable variables.

Numerical Results and Validation

The presented methodology demonstrates robust performance across several benchmark models. For instance, the proposed techniques significantly reduce the complexity and time required for performing structural identifiability tests, suggesting their substantial potential for broader application across scientific and engineering domains.

Implications and Future Work

The implications of this research extend to various dynamical system analysis applications, such as model linearization, parameter estimation, and control design. The authors propose that the development of advanced variable replacement heuristics could further enhance the efficacy of the projection-based elimination. Beyond immediate applications, the paper points toward future research exploring enhanced geometry-based bounds for differential systems and integrating probabilistic testing approaches for expanded model classes.

The availability of the implementation as a Julia package within the SciML ecosystem is poised to facilitate adoption and integration into scientific workflows, providing a powerful tool for researchers tackling complex dynamical models.

In summary, the paper delivers a substantial contribution to the field of differential algebraic elimination and structural identifiability, promising to advance both theoretical understanding and practical capabilities in the analysis of dynamical systems. Future work is anticipated to further refine these methods and broaden their applicability, potentially underpinned by the research community's continued engagement with the available software implementations.