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Signature-Based Universal Bilinear Approximations for Nonlinear Systems and Model Order Reduction

Martin Redmann, Justus Werner

TL;DR

The paper presents a signature-based universal bilinear surrogate for nonlinear controlled systems, enabling data-driven approximation of outputs via $y \approx C S$ with $S$ evolving bilinearly as $\dot{S}=A_0 S+\sum_{i=1}^m A_i S u_i(t)$. It establishes a rigorous connection between path signatures and control, proves a universal approximation property for maps of the control, and develops a time-limited MOR framework to reduce high-dimensional signature models even when the underlying nonlinear dynamics are unstable or nonzero-initialized. The approach learns only the output-projection matrix $C$ from data, uses a collapsed bilinear surrogate for efficient modeling and data fitting, and demonstrates scalability on a high-dimensional, non-Lipschitz reaction-diffusion example. Overall, the work extends MOR to signature-driven bilinear systems, enabling compact, accurate representations of complex nonlinear dynamics from observational data.

Abstract

This paper deals with non-Lipschitz nonlinear systems. Such systems can be approximated by a linear map of so-called signatures, which play a crucial role in the theory of rough paths and can be interpreted as collections of iterated integrals involving the control process. As a consequence, we identify a universal bilinear system, solved by the signature, that can approximate the state or output of the original nonlinear dynamics arbitrarily well. In contrast to other (bi)linearization techniques, the signature approach remains feasible in large-scale settings, as the dimension of the associated bilinear system grows only with the number of inputs. However, the signature model is typically of high order, requiring an optimization process based on model order reduction (MOR). We derive an MOR method for unstable bilinear systems with non-zero initial states and apply it to the signature, yielding a potentially low-dimensional bilinear model. An advantage of our method is that the original nonlinear system need not be known explicitly, since only data are required to learn the linear map of the signature. The subsequent MOR procedure is model-oriented and specifically designed for the signature process. Consequently, this work has two main applications: (1) efficient modeling/data fitting using small-scale bilinear systems, and (2) MOR for nonlinear systems. We illustrate the effectiveness of our approach in the second application through numerical experiments.

Signature-Based Universal Bilinear Approximations for Nonlinear Systems and Model Order Reduction

TL;DR

The paper presents a signature-based universal bilinear surrogate for nonlinear controlled systems, enabling data-driven approximation of outputs via with evolving bilinearly as . It establishes a rigorous connection between path signatures and control, proves a universal approximation property for maps of the control, and develops a time-limited MOR framework to reduce high-dimensional signature models even when the underlying nonlinear dynamics are unstable or nonzero-initialized. The approach learns only the output-projection matrix from data, uses a collapsed bilinear surrogate for efficient modeling and data fitting, and demonstrates scalability on a high-dimensional, non-Lipschitz reaction-diffusion example. Overall, the work extends MOR to signature-driven bilinear systems, enabling compact, accurate representations of complex nonlinear dynamics from observational data.

Abstract

This paper deals with non-Lipschitz nonlinear systems. Such systems can be approximated by a linear map of so-called signatures, which play a crucial role in the theory of rough paths and can be interpreted as collections of iterated integrals involving the control process. As a consequence, we identify a universal bilinear system, solved by the signature, that can approximate the state or output of the original nonlinear dynamics arbitrarily well. In contrast to other (bi)linearization techniques, the signature approach remains feasible in large-scale settings, as the dimension of the associated bilinear system grows only with the number of inputs. However, the signature model is typically of high order, requiring an optimization process based on model order reduction (MOR). We derive an MOR method for unstable bilinear systems with non-zero initial states and apply it to the signature, yielding a potentially low-dimensional bilinear model. An advantage of our method is that the original nonlinear system need not be known explicitly, since only data are required to learn the linear map of the signature. The subsequent MOR procedure is model-oriented and specifically designed for the signature process. Consequently, this work has two main applications: (1) efficient modeling/data fitting using small-scale bilinear systems, and (2) MOR for nonlinear systems. We illustrate the effectiveness of our approach in the second application through numerical experiments.

Paper Structure

This paper contains 18 sections, 17 theorems, 112 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Existing results and comparison to the signature approach
  3. Framework and story in a nutshell
  4. Foundations and signatures
  5. Signature approximations of continuous maps
  6. Locally Lipschitz continuous maps of the control process $u$
  7. Universal approximations by the signature
  8. MOR for unstable bilinear systems with non-zero initial states
  9. Identifying unimportant directions
  10. Reduced order model by time-limited balanced truncation
  11. Computation of Gramians
  12. Numerical experiments
  13. Learning the matrix $C$
  14. Approximating high-dimensional nonlinear systems
  15. Proofs of Section \ref{['3_1']}
  16. ...and 3 more sections

Key Result

Lemma 2.2

Let $u \in L^1([0,T];\mathbb{R}^m)$ and $U(\cdot)=\int_{0}^{\cdot} u(t) \, \mathrm{d}t$. The absolutely continuous function $U$ is of bounded variation. More precisely, it holds that

Figures (3)

  • Figure 1: Three example for $y(\cdot, u)$ in \ref{['output_num']} with $u= u^{(1)}, u^{(10)}, u^{(50)}$.
  • Figure 2: Error signature approximation of $y$ in \ref{['output_num']} (solid line) and error when reducing the dimension of the signature (circles).
  • Figure 3: First $50$ Hankel singular singular values of the signature model approximating the output in \ref{['output_num']}.

Theorems & Definitions (47)

  • Definition 2.1: Bounded Variation
  • Lemma 2.2
  • proof
  • Lemma 2.3: Natanson
  • Definition 2.4: (Truncated) Signature
  • Example 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Lemma 2.8: Uniqueness of the Signature
  • ...and 37 more