# Using Signatures and Koopman Operator to Learn Nonlinear Dynamics

Stéphane Chrétien, Ben Gao, Jordan Patracone, Olivier Alata

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Stéphane Chrétien, Ben Gao, Jordan Patracone, Olivier Alata. Using Signatures and Koopman Operator to Learn Nonlinear Dynamics. GSI'25 - 7th International Conference on Geometric Science of Information, Oct 2025, saint-malo, France. hal-05224359

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# Using Signatures and Koopman Operator to
Learn Nonlinear Dynamics

Stéphane Chrétien<fnref n="1" />, Ben Gao<fnref n="2" />,<fnref n="3" />, Jordan Patracone<fnref n="2" />,<fnref n="3" /> and Olivier Alata<fnref n="3" />

**Abstract.** We propose a novel framework for predicting the evolution of dynamical systems by learning the Koopman operator in the space of linear functionals on the Signature transform of trajectory data. The Signature, a central object in rough path theory, provides a universal and compact representation of paths through iterated integrals, enabling linear models to approximate a wide class of non-linear functionals. By restricting observables to lie in the span of truncated Signatures, we construct a finite-dimensional approximation of the Koopman operator, which we estimate directly from data using regularized linear regression. This approach merges the expressiveness of operator-theoretic methods with the structural richness of Signature features.

**Keywords:** Koopman operator approximation · Signature transform · Non-linear dynamical systems.

## 1 Introduction

Understanding and predicting the evolution of dynamical systems is a central problem in many fields, from physics [6] and engineering [1] to finance [11] and neuroscience [2]. Traditional approaches typically rely on identifying explicit governing equations or employing data-driven models such as autoregressive schemes or neural networks. These methods, while powerful, often struggle to generalize across systems with complex, non-linear, or high-dimensional dynamics, particularly when the system structure is only partially observed. A promising alternative lies in operator-theoretic approaches to dynamics, particularly through the lens of the Koopman operator [4,15]. The Koopman operator acts on observables of the system rather than the state itself, lifting the non-linear evolution of states into a (typically infinite-dimensional) linear evolution in function space. This linearity opens the door to new learning paradigms in which the focus shifts from modeling states directly to learning how observables evolve under the system's dynamics.

<footnote n="1">¹ Université Lyon 2, Laboratoire ERIC, Bron, France
stephane.chretien@univ-lyon2.fr</footnote>

<footnote n="2">² Inria</footnote>

<footnote n="3">³ Université Jean Monnet Saint-Etienne, CNRS, Institut d'Optique Graduate School,
Laboratoire Hubert Curien UMR 5516, F-42023, SAINT-ETIENNE, France
ben.gao@univ-st-etienne.fr</footnote>

<footnote>jordan.patracone@univ-st-etienne.fr</footnote>

<footnote>olivier.alata@univ-st-etienne.fr</footnote>
In this work, we propose a framework that leverages the *Signature transform* [5]—a core object in rough path theory—as a basis for representing system trajectories. The Signature provides a graded, algebraic summary of a path in terms of iterated integrals, and crucially, the space of linear functionals on Sig-natures is dense in the space of continuous functionals on paths. This makes the Signature transform an ideal candidate for learning in the function space where the Koopman operator acts.

Empirical results on benchmark dynamical systems demonstrate that our method effectively captures the temporal evolution and generalizes well beyond the training data. Our contribution thus lies at the intersection of signature-based learning and operator-theoretic dynamics, offering a principled and efficient framework for forecasting complex time series.

# 2 Background on Koopman Theory

Koopman operator theory offers a linear perspective on non-linear dynamical systems by focusing on the evolution of *observables*—complex-valued functions defined on the state space—rather than the evolution of the state itself. The Koopman operator was originally introduced by B.O. Koopman and Von Neumann [7,13] for ergodic systems. The renewed interest in the Koopman operator is due to previous theoretical works [12,8] that enable the application of spectral theory to study complex, non-linear dynamical systems and the potential for global linearization.

## 2.1 Koopman operator and Spectral properties

Let $\mathcal{X} \subset \mathbb{R}^d$ be the state space, and consider a discrete dynamical system governed by the map $T: \mathcal{X} \to \mathcal{X}$:

$$x_{t+1} = Tx_t. \qquad (1)$$

For a space of complex-valued observables $\mathcal{F} \subset L^2(\mathcal{X}, \mu)$, the Koopman operator $\mathcal{K}: \mathcal{F} \to \mathcal{F}$ is defined by:

$$(\mathcal{K}f)(x) = f(Tx), \qquad (2)$$

where $f \in \mathcal{F}$ is called an observable. This operator is linear, even when $T$ is non-linear. As such, the evolution of non-linear dynamics can be studied through the spectral properties of this infinite-dimensional linear operator. Its spectral decomposition provides insight into the long-term behaviour of the system. If $\phi \in \mathcal{F}$ is an eigenfunction of $\mathcal{K}$ with eigenvalue $\mu \in \mathbb{C}$, then:

$$\mathcal{K}\phi = \mu\phi, \qquad (3)$$

and the evolution of $\phi$ over time is simply governed by exponential scaling. The spectrum of $\mathcal{K}$ can include a combination of point spectrum (eigenvalues)

$$\sigma_p(\mathcal{K}) = \{\lambda : \mathcal{K} - \lambda I \text{ is not injective}\}, \qquad (4)$$
continuous spectrum, and residual spectrum, depending on the regularity and mixing properties of the system and the function space $\mathcal{F}$ [15]. If one can find $L$ eigenpairs $\{(\lambda_1, \phi_1), (\lambda_2, \phi_2), \dots, (\lambda_L, \phi_L)\}$, such that the change of coordinates

$$x \mapsto \phi_1(x), \phi_2(x), \dots, \phi_L(x) \qquad (5)$$

is injectif, then the dynamical system can be globally linearized.

## 2.2 Data-driven finite approximation and Time delay coordinates

In practice, the Koopman operator cannot be represented explicitly due to its infinite dimensionality. Numerical methods aim to approximate it by projecting onto a finite-dimensional subspace spanned by a chosen set of basis functions $\{\psi_1, \dots, \psi_N\}$. These functions can be fixed chosen carefully the popular algorithm is Extended Dynamic Mode Decomposition (EDMD) [16], which approximates $\mathcal{K}$ by solving a least-squares problem. Given snapshots $\{x_t, x'_t\}_{t=1}^M$, where $x'_t = Tx_t$, and defining feature vectors

$$\Psi(x) = [\psi_1(x), \dots, \psi_N(x)]^\top \qquad (6)$$

EDMD computes the best-fit linear operator $\mathbf{K} \in \mathbb{C}^{N \times N}$ such that:

$$\Psi(x'_t) \approx \mathbf{K}\Psi(x_t), \quad \text{for } t = 1, \dots, T. \qquad (7)$$

This operator $\mathbf{K}$ serves as a finite-dimensional approximation to $\mathcal{K}$ associated with the functions that where chosen to span the finite dimensional space where the dynamics is restricted to be observed. The eigenvalues and eigenvectors of $\mathbf{K}$ then approximate the spectral properties of $\mathcal{K}$. Note that the basis functions can also be learnable [9,14].

The time delay coordinates are also used as Koopman embeddings [3], provided that the conditions of Takens embedding theorem are satisfied. The state $x_t$ at time $t$ is augmented to

$$\mathbf{x}_t^{(w)} = [x_{t-w}, x_{t-w+1}, \dots, x_t]^{\top}. \qquad (8)$$

Due to the partiality of observations and their discrete nature, the time delay coordinates often give superior performance in practice.

## 3 Learning the Koopman operator in Signature space

In our approach, we take the observable space to be the space of linear functionals on the Signature transform of segments of trajectories (time delay coordinates). The Signature provides a graded, algebraically structured representation of paths in terms of iterated integrals, and it enjoys a universal approximation property: any continuous functional on path space can be approximated arbitrarily well by linear functionals of its Signature [10].
## 3.1 Signature transform

Let $\gamma: [0, T] \to \mathbb{R}^d$ denote a path representing a segment of a dynamical system trajectory. The Signature of $\gamma$, denoted $S(\gamma)$, is the collection of all iterated integrals of $\gamma$, formally given by:

$$S(\gamma) = \left( 1, \int_{0 \le t_1 \le T} d\gamma_{t_1}, \int_{0 \le t_1 \le t_2 \le T} d\gamma_{t_1} \otimes d\gamma_{t_2}, \dots \right) \in T((\mathbb{R}^d)) \quad (9)$$

where $T((\mathbb{R}^d))$ is the tensor algebra over $\mathbb{R}^d$. In practice, we work with a finite truncation of the Signature up to level $m$, denoted $S^{\le m}(\gamma) \in \mathbb{R}^N$, with $N = \sum_{k=0}^m d^k$.

In practice, we often work with a finite truncation of the Signature up to level $m$, denoted by

$$S^{\le m}(\gamma) = \left( 1, \int_{0 \le t_1 \le T} d\gamma_{t_1}, \int_{0 \le t_1 \le t_2 \le T} d\gamma_{t_1} \otimes d\gamma_{t_2}, \dots, \int_{0 \le t_1 \le \dots \le t_m \le T} d\gamma_{t_1} \otimes \dots \otimes d\gamma_{t_m} \right) \in \mathbb{R}^N \quad (10)$$

where the dimension of this truncated signature space is $N = \sum_{k=0}^m d^k$. This truncation is necessary for computational purposes since the full signature is an infinite series. The truncated signature $S^{\le m}(\gamma)$ provides a rich, hierarchical summary of the path's geometric properties up to complexity level $m$. The readers are referred to [5] for a thorough introduction. In practice, given an observed discrete trajectory $\mathbf{x}_T = (x_0, x_1, \dots, x_T)$, we calculate truncated signatures on its segments that correspond to time delay coordinates $\mathbf{x}_t^{(w)}$.

## 3.2 Lyons' Universality Theorem

The universality of the Signature is a powerful result that connects the algebraic structure of paths to the approximation of functions defined on those paths.

**Theorem 1.** [10] Let $\mathcal{P}$ be a compact set of bounded variation paths $\gamma: [0, T] \to \mathbb{R}^d$. Then for any continuous functional $F: \mathcal{P} \to \mathbb{R}$, there exists a linear functional $\ell \in T((\mathbb{R}^d))^*$ such that:

$$\sup_{\gamma \in \mathcal{P}} |F(\gamma) - \langle \ell, S(\gamma) \rangle| < \varepsilon, \quad (11)$$

for any $\varepsilon > 0$. That is, linear functionals of the Signature are dense in the space of continuous functionals on path space.

This means that Signatures form a universal feature set: any continuous observable of a path can be approximated arbitrarily well by a linear combination of Signature terms. The proof of this theorem is based on the following properties:
- Tensor algebra: The Signature $S(\gamma)$ lives in the tensor algebra $T((\mathbb{R}^d))$, an infinite series $(1, S^1(\gamma), S^2(\gamma),...)$, where each $S^k(\gamma) \in (\mathbb{R}^d)^{\otimes k}$ is the k-fold iterated integral of $\gamma$.

- Algebra of functionals: Linear functionals $\ell \in T((\mathbb{R}^d))^*$ act on these tensor terms to extract features. For example, the functional associated with a multi-index $(i_1, i_2)$ maps to the iterated integral $\int \gamma_{i_1} d\gamma_{i_2}$.

- Topology: The space $\mathcal{P}$ of bounded variation paths is equipped with the 1-variation norm, making it a Banach space. The continuity of $F$ is with respect to this topology. Under these settings, Lyons' theorem guarantees that any $F \in C(\mathcal{P})$ can be approximated in the uniform norm by linear functionals on the Signature.

**Implication for learning** Lyon's theorem gives theoretical justification for using truncated Signatures as feature vectors for learning tasks on path-valued data. Specifically:

1. Feature sufficiency: The Signature contains enough information to approximate any continuous function of a path. This makes it a complete representation for learning, in contrast to heuristic feature maps.

2. Linear models are sufficient: Once the Signature is used as a feature map, any learning problem (e.g., regression, classification, or operator estimation) becomes linear in the feature space. Non-linear tasks in path space become linear in Signature space.

Given that linear functionals on the Signature are dense in the space of continuous functionals on path space, we propose to represent observables $f$ as linear functionals

$$f(\gamma) \approx \langle w, S^{\leq m}(\gamma) \rangle, \quad w \in \mathbb{R}^N. \qquad (12)$$

for $m$ sufficiently large. This representation enables us to restrict the Koopman operator $\mathcal{K}$ to a finite-dimensional subspace $\mathcal{F}_m \subset C(\mathcal{X}^T)$ consisting of such linear functionals on Signatures.

## 3.3 Finite-dimensional approximation

Let $\Phi: \gamma \mapsto S^{\leq m}(\gamma) \in \mathbb{R}^N$ denote the Signature feature map. Our goal is to estimate a matrix $K \in \mathbb{R}^{N \times N}$ that approximates the action of $\mathcal{K}$ in this feature space:

$$\mathcal{K}f(\gamma) \approx \langle w, K\Phi(\gamma) \rangle, \quad \forall f(\gamma) = \langle w, \Phi(\gamma) \rangle. \qquad (13)$$

In other words, $K$ is a linear operator acting on the feature space $\mathbb{R}^N$ such that the predicted value of an observable at the next time step is:

$$f(\gamma') \approx \langle w, K\Phi(\gamma') \rangle, \qquad (14)$$

where $\gamma'$ denotes the future trajectory segment (possibly of the same length and resolution as $\gamma$) overlapping.
## 3.4 Data-driven estimation

To estimate $K$ from data, we assume access to a dataset $\{(\gamma_i, \gamma'_i)\}_{i=1}^M$ consisting of overlapping trajectory segment pairs from a dynamical system, where $\gamma_i$ represents a past path segment and $\gamma'_i$ its immediate future. Define:

- $X \in \mathbb{R}^{M \times N}$, where each row $X_i = \Phi(\gamma_i)$ is the Signature of a past segment.

- $Y \in \mathbb{R}^{M \times N}$, where $Y_i = \Phi(\gamma'_i)$ is the Signature of the corresponding future trajectory.

We solve the following least-squares problem:

$$K^* = \arg \min_{K \in \mathbb{R}^{N \times N}} \|Y - XK^\top\|_F^2 + \lambda \|K\|_F^2, \quad (15)$$

where $\|\cdot\|_F$ denotes the Frobenius norm, and $\lambda \ge 0$ is a regularization parameter. The optimal solution has a closed-form expression:

$$K^* = (X^\top X + \lambda I)^{-1} X^\top Y. \quad (16)$$

This yields a linear approximation of the Koopman operator restricted to the space of linear functionals on truncated Signatures.

**Theoretical setup** We consider path segments $\gamma \in \mathcal{X}^T$ from a dynamical system and apply the truncated Signature transform $\Phi(\gamma) := S^{\le m}(\gamma) \in \mathbb{R}^N$. The finite Koopman operator $K^*$ is defined by minimizing the expected squared error:

$$K^* := \arg \min_K \mathbb{E} \left[ \| \Phi(\gamma') - K\Phi(\gamma) \|^2 \right]. \quad (17)$$

Given $M$ samples $\{(\gamma_i, \gamma'_i)\}_{i=1}^M$, we define:

$$X = \begin{bmatrix} \Phi(\gamma_1)^\top \\ \vdots \\ \Phi(\gamma_M)^\top \end{bmatrix}, \quad Y = \begin{bmatrix} \Phi(\gamma'_1)^\top \\ \vdots \\ \Phi(\gamma'_M)^\top \end{bmatrix}. \quad (18)$$

The empirical Koopman estimator is:

$$\hat{K}_{\lambda} = (X^{\top} X + \lambda I)^{-1} X^{\top} Y. \quad (19)$$

**Future state prediction** One needs to invert signatures to go back to state space. If reconstructing the segment from its truncated signature remains impractical, we show that, by leveraging the first level of the signature, we can easily predict future states of dynamics.

For a new segment $\gamma_{new} = [x_{t-w}, x_{t-w+1}, \dots, x_t]^\top$, we seek to predict $\gamma'_{new} = [x_{t-w+1}, x_{t-w+2}, \dots, x_{t+1}]^\top$, where $x_{t+1}$ is the only unknown state.
Since the first level signature represents the increments of each dimension of the original path segment, we have

$$ \begin{bmatrix} x_{t+1}^1 \\ \vdots \\ x_{t+1}^d \end{bmatrix} = \begin{bmatrix} x_{t-w+1}^1 \\ \vdots \\ x_{t-w+1}^d \end{bmatrix} + \begin{bmatrix} S(\gamma_{new}')^1 \\ \vdots \\ S(\gamma_{new}')^d \end{bmatrix}. \quad (20) $$

Therefore, the prediction $\hat{x}_{t+1}$ is given by using $\hat{K}_{\lambda}\Phi(\gamma_{new})$ that approximates $S(\gamma_{new}')$.

# 4 Experiments

We test our method on simulated data of two dynamic systems governed by known differential equations. The results show that our method learns a good approximation of $\mathcal{K}$ and generalizes well on test data. Our implementation is available at https://github.com/ben2022lo/SigKoop.

## 4.1 Single attractor

The first dynamic system has one single fixed point [4]

$$ \begin{aligned} \dot{u} &= \mu u \\ \dot{v} &= b(v - u^2) \end{aligned} \quad (21) $$

<figure><img src="image_2.png" /><figcaption>Fig. 1: Prediction(left) vs Ground-truth(right) for test trajectories</figcaption></figure>

Taking $\mu = -0.05$ and $b = -1$, the attracting manifold is given by $v = u^2$. We simulate 1000 trajectories of 100 time steps with $\Delta t = 0.01$ by randomly generating initial states on the domain $[-2, 2] \times [-2, 2]$. The train-test split ratio is 0.8. The Signature features are computed up to level 3. We evaluate the performance by Mean Squared Error applied to the state variable. The train error and the test error are both of order $10^{-7}$. We show ground truth and prediction for test trajectories in Figure 1. Figure 2 illustrates the distributions of trajectory-wise error.
<figure><img src="image_3.png" /><figcaption>Fig. 2: Trajectory-error distribution Train(left) vs Test(right)</figcaption></figure>

## 4.2 Duffing oscillator

The second dynamic is governed by a Duffing equation

$$ \ddot{x} = -\delta\dot{x} - x(\beta + \alpha x^2), \qquad (22) $$

with $\delta = 0.5, \beta = -1, \alpha = 1$, exhibiting two stable spirals at $x = \pm 1, \dot{x} = 0$ and a saddle point at $x = 0, \dot{x} = 0$ [16].

<figure><img src="image_4.png" /><figcaption>Fig. 3: Prediction(left) vs Ground-truth(right) for test trajectories</figcaption></figure>

The training setting is the same as for the first dynamic, except that we take $\Delta t = 0.025$ to avoid convergence of trajectories to fixed points at first time steps. The training error is $6.54 \times 10^{-6}$ and the test error is $6.15 \times 10^{-6}$. Figure 3 and 4 show that our method allows generalization to test trajectories.

## 5 Conclusion

In this work, we proposed a novel method for approximating the Koopman operator by leveraging the expressive power of the Signature transform. This approach
<figure><img src="image_5.png" /><figcaption>Fig. 4: Trajectory-error distribution Train (left) vs Test (right)</figcaption></figure>

combines the universality of Signatures for representing path-dependent functionals with the linear structure of Koopman operator theory. Empirical results on benchmark systems, including a simple attractor and the Duffing oscillator, show that our method generalizes well and achieves low prediction error. In future work, we plan to deepen the theoretical analysis of the Koopman estimator learned in Signature space, particularly regarding convergence rates compared to other choices of Koopman embeddings

**Acknowledgements** This work was sponsored by a public grant overseen by Auvergne-Rhône-Alpes region, Grenoble Alpes Metropole and BPIFrance, as part of project I-Démo Région "Green AI".

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