Periodic Reporting for period 4 - Back to the Roots (Back to the roots of data-driven dynamical system identification)
Periodo di rendicontazione: 2025-07-01 al 2025-12-31
Modern technologies increasingly rely on mathematical models that learn from data in fields such as manufacturing, autonomous driving and medical monitoring. However, many of today’s widely used modeling techniques depend on optimization methods that may produce different answers each time they are run. This lack of transparency and reproducibility is a growing concern as society depends more and more on data-driven AI technologies.
Our project set out to create a rigorous and trustworthy mathematical framework for learning dynamic models from data—a task known as system identification. To achieve this, we pursued three objectives, moving from fundamental mathematical insights to practical algorithms and software that others can use.
A key breakthrough was solving a long-standing open problem: we proved that, for a broad class of dynamic models, the best model can be found via an eigenvalue problem—a technique that is well understood by engineers. This required combining ideas from several branches of mathematics.
Below, we give a concise overview of the three objectives and the key results we achieved.
OBJECTIVE 1: UNDERSTANDING THE MATHEMATICS BEHIND DATA-DRIVEN MODELS
We began by studying the mathematical structures that underlie the equations appearing in data-driven modeling. We clarified how the so-called fundamental subspaces of (block-)Macaulay matrices capture the relationships between data, equations and solutions. These insights allowed us to solve difficult algebraic problems such as multivariate polynomial equations and multiparameter eigenvalue problems (MEPs).
OBJECTIVE 2: OPTIMAL MODELS FROM NON-MODEL-COMPLIANT DATA
Next, we addressed the notoriously hard problem of constructing the best possible model when the available data do not perfectly follow the model equations. We introduced a new framework that reformulates this task as an MEP, which makes it possible to compute models that are provably optimal. The same idea also yields optimal reduced-order models, an important tool for simplifying complex systems.
OBJECTIVE 3: EFFICIENT ALGORITHMS FOR OPTIMAL MODELS
Finally, we engineered new numerical algorithms capable of solving the polynomial problems and thereby delivering the optimal solutions to identification tasks. All methods were integrated into our publicly available software toolbox MacaulayLab, enabling researchers and practitioners to apply our results directly in their own modeling tasks.
Our project set out to create a rigorous and trustworthy mathematical framework for learning dynamic models from data—a task known as system identification. To achieve this, we pursued three objectives, moving from fundamental mathematical insights to practical algorithms and software that others can use.
A key breakthrough was solving a long-standing open problem: we proved that, for a broad class of dynamic models, the best model can be found via an eigenvalue problem—a technique that is well understood by engineers. This required combining ideas from several branches of mathematics.
Below, we give a concise overview of the three objectives and the key results we achieved.
OBJECTIVE 1: UNDERSTANDING THE MATHEMATICS BEHIND DATA-DRIVEN MODELS
We began by studying the mathematical structures that underlie the equations appearing in data-driven modeling. We clarified how the so-called fundamental subspaces of (block-)Macaulay matrices capture the relationships between data, equations and solutions. These insights allowed us to solve difficult algebraic problems such as multivariate polynomial equations and multiparameter eigenvalue problems (MEPs).
OBJECTIVE 2: OPTIMAL MODELS FROM NON-MODEL-COMPLIANT DATA
Next, we addressed the notoriously hard problem of constructing the best possible model when the available data do not perfectly follow the model equations. We introduced a new framework that reformulates this task as an MEP, which makes it possible to compute models that are provably optimal. The same idea also yields optimal reduced-order models, an important tool for simplifying complex systems.
OBJECTIVE 3: EFFICIENT ALGORITHMS FOR OPTIMAL MODELS
Finally, we engineered new numerical algorithms capable of solving the polynomial problems and thereby delivering the optimal solutions to identification tasks. All methods were integrated into our publicly available software toolbox MacaulayLab, enabling researchers and practitioners to apply our results directly in their own modeling tasks.
Throughout the project, we carried out the work defined in the three objectives and translated our mathematical insights into practical tools for researchers and engineers. The overview below provides somewhat more technical detail than the previous summary.
OBJECTIVE 1: UNDERSTANDING THE MATHEMATICS BEHIND DATA-DRIVEN MODELS
We revealed how the four fundamental subspaces of (block-)Macaulay matrices provide a unified framework for solving the core tasks in our modeling approach—multivariate polynomial equations and multiparameter eigenvalue problems (MEPs). The right null space shows that the solutions can be computed as the eigenvalues of an associated matrix; the left null space describes how the equations interact; the row space delivers the difference equations that define multi dimensional (mD) systems and enables numerical elimination in both polynomial equations and polynomial dynamic models; the properties of the column space can be used to speed up calculations.
OBJECTIVE 2: OPTIMAL MODELS FROM NON-MODEL-COMPLIANT DATA
We proved that globally optimal models can be obtained by formulating the optimization problem as a rectangular MEP. The same approach yields a globally optimal model reduction method. Working with the associated MEPs also enabled us, for the first time, to count the exact number of solutions.
OBJECTIVE 3: EFFICIENT ALGORITHMS FOR OPTIMAL MODELS
We developed efficient numerical algorithms exploiting the special structure and properties of the underlying matrices. A key outcome is MacaulayLab (http://macaulaylab.net(si apre in una nuova finestra)) a publicly available MATLAB toolbox for solving multivariate polynomial equations and MEPs, including a database of test problems.
Furthermore, we introduced two elimination methods based on tools from numerical linear algebra, such as the well-known singular value decomposition (SVD) and the less well known cosine–sine decomposition (CSD). We also developed an SVD‑based state elimination technique for polynomial dynamic models.
EXPLOITATION AND DISSEMINATION
We disseminated our results through peer reviewed publications, conferences, seminars and invited talks, listed at https://homes.esat.kuleuven.be/~sistawww/bdm/backtotheroots/(si apre in una nuova finestra).
MacaulayLab makes our algorithms widely accessible to researchers and practitioners.
The freely available draft of our didactic book “Back to the Roots” presents the framework to a broad mathematical‑engineering audience and supports adoption in research and teaching.
OBJECTIVE 1: UNDERSTANDING THE MATHEMATICS BEHIND DATA-DRIVEN MODELS
We revealed how the four fundamental subspaces of (block-)Macaulay matrices provide a unified framework for solving the core tasks in our modeling approach—multivariate polynomial equations and multiparameter eigenvalue problems (MEPs). The right null space shows that the solutions can be computed as the eigenvalues of an associated matrix; the left null space describes how the equations interact; the row space delivers the difference equations that define multi dimensional (mD) systems and enables numerical elimination in both polynomial equations and polynomial dynamic models; the properties of the column space can be used to speed up calculations.
OBJECTIVE 2: OPTIMAL MODELS FROM NON-MODEL-COMPLIANT DATA
We proved that globally optimal models can be obtained by formulating the optimization problem as a rectangular MEP. The same approach yields a globally optimal model reduction method. Working with the associated MEPs also enabled us, for the first time, to count the exact number of solutions.
OBJECTIVE 3: EFFICIENT ALGORITHMS FOR OPTIMAL MODELS
We developed efficient numerical algorithms exploiting the special structure and properties of the underlying matrices. A key outcome is MacaulayLab (http://macaulaylab.net(si apre in una nuova finestra)) a publicly available MATLAB toolbox for solving multivariate polynomial equations and MEPs, including a database of test problems.
Furthermore, we introduced two elimination methods based on tools from numerical linear algebra, such as the well-known singular value decomposition (SVD) and the less well known cosine–sine decomposition (CSD). We also developed an SVD‑based state elimination technique for polynomial dynamic models.
EXPLOITATION AND DISSEMINATION
We disseminated our results through peer reviewed publications, conferences, seminars and invited talks, listed at https://homes.esat.kuleuven.be/~sistawww/bdm/backtotheroots/(si apre in una nuova finestra).
MacaulayLab makes our algorithms widely accessible to researchers and practitioners.
The freely available draft of our didactic book “Back to the Roots” presents the framework to a broad mathematical‑engineering audience and supports adoption in research and teaching.
This project advanced the state of the art in system identification, model reduction, and polynomial computation by turning long‑standing optimization problems into solvable multiparameter eigenvalue problems; by determining, for the first time, the exact number of solutions these problems admit; and by developing efficient numerical algorithms that are now available to the community through our software.
We highlight the key advances achieved in this project.
• We strengthened the foundations of polynomial equation solving by developing a Macaulay‑matrix method that operates directly on its column space, avoiding the need to compute the right null space.
• We created the first numerical linear‑algebra algorithms able to handle rectangular multiparameter eigenvalue problems (MEPs), a class of problems that had no dedicated solvers before this project.
• Delivering MacaulayLab required a substantial engineering effort far beyond implementing algorithms: we redesigned core routines for efficiency and memory use, built extensive tests and benchmarks, and invested in thorough documentation and illustrative examples—ensuring that researchers and engineers can rely on it in practice.
• For least‑squares realization and H2 model reduction, we introduced deterministic, globally optimal methods. For LS realization, we established the first-ever approach that computes all local and global minimizers and revealed the double‑FIR structure of optimal misfits; for model reduction, we derived globally optimal reduced‑order models obtained through an eigenvalue‑based formulation.
• Most notably, we obtained an explicit, first‑of‑its‑kind count of all complex critical points in both settings—a genuine breakthrough made possible only by combining algebraic‑geometry tools with insights from system theory and numerical linear algebra.
• We extended these ideas to multidimensional (mD) systems, deriving their defining difference equations, clarifying the structure of their local minima, and providing unified kernel, image and state‑space descriptions.
• We introduced new SVD-based methods for eliminating variables in polynomial equations and states in polynomial dynamic models. When applied to polynomial optimization problems, these elimination techniques yield the critical value polynomial, providing a univariate encoding of all stationary objective values and implicit relations for parametric optimization problems.
We highlight the key advances achieved in this project.
• We strengthened the foundations of polynomial equation solving by developing a Macaulay‑matrix method that operates directly on its column space, avoiding the need to compute the right null space.
• We created the first numerical linear‑algebra algorithms able to handle rectangular multiparameter eigenvalue problems (MEPs), a class of problems that had no dedicated solvers before this project.
• Delivering MacaulayLab required a substantial engineering effort far beyond implementing algorithms: we redesigned core routines for efficiency and memory use, built extensive tests and benchmarks, and invested in thorough documentation and illustrative examples—ensuring that researchers and engineers can rely on it in practice.
• For least‑squares realization and H2 model reduction, we introduced deterministic, globally optimal methods. For LS realization, we established the first-ever approach that computes all local and global minimizers and revealed the double‑FIR structure of optimal misfits; for model reduction, we derived globally optimal reduced‑order models obtained through an eigenvalue‑based formulation.
• Most notably, we obtained an explicit, first‑of‑its‑kind count of all complex critical points in both settings—a genuine breakthrough made possible only by combining algebraic‑geometry tools with insights from system theory and numerical linear algebra.
• We extended these ideas to multidimensional (mD) systems, deriving their defining difference equations, clarifying the structure of their local minima, and providing unified kernel, image and state‑space descriptions.
• We introduced new SVD-based methods for eliminating variables in polynomial equations and states in polynomial dynamic models. When applied to polynomial optimization problems, these elimination techniques yield the critical value polynomial, providing a univariate encoding of all stationary objective values and implicit relations for parametric optimization problems.