Trondheim. Learning Differential Equations via Chen–Fliess series

This article presents an accessible overview of how Chen–Fliess series can be combined with modern learning techniques to model and analyze nonlinear dynamical systems. The contribution fits well within ECMI’s mission of connecting advanced mathematics with applications in science and engineering.

by Kurusch Ebrahimi-Fard and W. Steven Gray

What Is a Chen–Fliess Series?

When mathematics, computer science, and engineering meet, interesting things can happen. Concepts that once lived in rather abstract corners of mathematics often reappear as practical tools for understanding real-world problems.

Chen–Fliess series are a beautiful example of this kind of cross-disciplinary phenomenon [7]. They arise in the study of dynamical systems driven by external inputs and provide a systematic way to describe how a system responds over time to an applied input. In applications, they can be used to characterize complex interconnected networks of dynamical systems like smart grids, neural networks, and population dynamics.

Paths, Words, and Memory

The story begins with Kuo-Tsai Chen, a mathematician who in the 1950s was interested in how
to describe the evolution of a path in an m-dimensional space. His insight was that a path carries memory and that different directions interact over time in
a noncommutative fashion [3, 4].

To capture this idea concretely, Chen introduced what are now called iterated integrals. Let $u(t)=(u_1(t), …, u_m(t))$ be a time dependent path in m dimensions. Chen’s iterated integrals encode the cumulative and ordered effects of each path component $u_i$. For example, the first-level integrals measure total accumulation,

while higher-level iterated integrals record interactions in time,

and so on. Taken together, all iterated integrals of all orders capture not only how much the path changes, but how different components influence each other over time. Around the same time that Chen was presenting his seminal work, a very different set of concepts was emerging in theoretical computer science. In automata theory and formal language theory, words, that is, finite sequences of symbols from an alphabet $A=\{1,\dots,m\}$, were being

used to describe logical processes, rules, and transitions [15]. Words became a way to encode structure and order. The connection between the two disciplines can be seen from the fact that an iterated integral like $E_{ij}[u]$ is uniquely identified by the word $ij$. In general, $ij \neq ji$ , and likewise, $E_{ij}[u]\neq E_{ji}[u]$.

Michel Fliess and Dynamical Systems

Michel Fliess was the first to realize that paths could be interpreted as inputs driving a dynamical system.
A large class of systems in control theory and engineering can be written as controlled ordinary differential equations of the form

together with an output function $y=h(z).$ Here the vector fields $f_i$ describe how the state evolves when driven by the input channels $u_i$. The central question is how to express the solution $z(t)$, or some output derived from it, in terms of the input $u$.

Fliess showed that the input–output map $u$→$y$ of such systems can be expanded in terms of linear combinations of Chen’s iterated integrals [8, 9]. The resulting expression is called a Chen–Fliess series . The coefficients of this series are computed directly in terms of the vector fields and the output function to yield a series of the form

where:
1. $w ={i_1} \cdots {i_k}$ runs over all words, i.e., finite sequences made from letters of the alphabet $A={0,1,\ldots,m}$ encoding ordered input channels ( $u_0:=1$ ),

2.
$E_w[u](t)$ denotes the iterated integral associated with the word $w,$

3. each real-valued coefficient $c_w$ depends on the system dynamics but is independent of the specific coordinate frame.

The series can be shown to converge when certain restrictions are imposed on the vector fields, the output function, and the size of the input.

Why This Matters

What makes Chen–Fliess series special is their unifying role. They bring together ideas from geometry (paths and trajectories), algebra (words and their combinatorics), dynamical systems (how states evolve over time), and engineering (system control and signal processing).
In particular, they provide a language of memory for dynamical systems. This viewpoint has proved to be powerful. It underlies modern approaches to nonlinear control, system interconnections, and feedback.
For example, the computational machinery provided by this language was used to describe how biological shunting neural networks in animals and insects process visual information to detect motion in their environment and estimate velocity [13]. More recently, closely related ideas have appeared in areas such as rough path theory and machine learning, where structured representations of time-dependent data play a central role [5].

Advanced Perspectives and the Future

The word-series nature of Chen-Fliess series provides rich algebraic structures which are
naturally described using Hopf algebras [2]. This viewpoint provides powerful
tools for understanding structural properties of interconnected nonlinear systems [6,11] and efficient computational schemes for nonlinear control [12], stochastic analysis [1,10] and machine learning [14]. As more powerful computing platforms like quantum computers become available, it is likely the role of Chen-Fliess series in
analysis and engineering design will continue to flourish.

References

[1] Peter Bank, Christian Bayer, Paul P.Hager, Sebastian Riedel, and Tobias Nauen. Stochastic control with signatures. SIAM J. Control Optim. , 63(5):3189–3218, 2025.

[2] Pierre Cartier and Frédéric Patras. Classical Hopf algebras and their applications , volume 29 of Algebra and Applications . Springer, Cham, 2021.

[3] Kuo-Tsai Chen. Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. of Math. (2) , 65:163–178, 1957.

[4] Kuo-TsaiChen. Integrationofpaths–afaithfulrepresentationofpathsbynon-commutative formal power series. Trans. Amer. Math. Soc. , 89:395–407, 1958.

[5] Ilya Chevyrev and Andrey Kormilitzin. A primer on the signature method in machine learning. In Signature Methods in Finance , Springer Finance, pages 3–64. Springer, Cham, 2026.

[6] Luis A. Duffaut Espinosa, Kurusch Ebrahimi-Fard, and W. Steven Gray. Combinatorial Hopf algebras for interconnected nonlinear input-output systems with a view towards discretization. In K. Ebrahimi-Fard and M. Barbero Liñán, editors, Discrete Mechanics, Geometric Integration and Lie-Butcher Series , volume 267 of Springer Proceedings in Mathematics & Statistics , pages 139–183. Springer International Publishing, Cham, Switzerland, 2018.

[7] Luis A. Duffaut Espinosa, Kurusch Ebrahimi-Fard, W. Steven Gray, and Venkatesh G. S. What is a… Chen–Fliess–Series? Notices Amer. Math. Soc. , 73(02):144–149, 2026.

[8] Michel Fliess. Fonctionnelles causales non linéaires et indéterminées non commutatives. Bull. Soc. Math. France, 109(1):3–40, 1981.

[9] Michel Fliess, Moustanir Lamnabhi, and Françoise Lamnabhi-Lagarrigue. An algebraic approach to nonlinear functional expansions. IEEE Trans. Circuits Syst. , 30(8):554–570, 1983.

[10] Mie Glückstad, Nicola Muca Cirone, and Josef Teichmann. Signature reconstruction from randomized signatures. https://arxiv.org/abs/2502.03163, 2025.

[11] W. Steven Gray, Luis A. Duffaut Espinosa, and Kurusch Ebrahimi-Fard. Faà di Bruno Hopf algebra of the output feedback group for multivariable Fliess operators. Systems Control Lett., 74:64–73, 2014.

[12] W. Steven Gray and Kurusch Ebrahimi-Fard. Generating series for networks of Chen–Fliess series.Systems Control Lett. , 147(1):article 104827, 2021.

[13] W. Steven Gray and Bahram Nabet. Volterra series analysis and synthesis of a neural network for velocity estimation. IEEE Trans. Syst., Man, Cybern. B. Cybern. , 29(2):190-197, 1999.

[14] Joshua Hanson and Maxim Raginsky. Rademacher complexity of neural ODEs via Chen-Fliess series. In Alessandro Abate, Mark Cannon, Kostas Margellos, and Antonis Papachristodoulou, editors, Proceedings of the 6th Annual Learning for Dynamics & Control Conference , volume 242 of Proceedings of Machine Learning Research , pages 758–769. PMLR, 15–17 Jul 2024.

[15] Marcel-Paul Schützenberger. On the definition of a family of automata. Inform. Control , 4:245–270, 1961.