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Linear identification of nonlinear systems: A lifting technique based on the Koopman operator
Alexandre Mauroy, Jorge Goncalves
TL;DR
The paper addresses identifying a nonlinear vector field from snapshot data by recasting the task as linear identification in observable space. It lifts data to identify a projected Koopman operator and connects that operator to the vector field; the authors report robustness to noise and recovery across different system behaviors.
Problem
The paper addresses identifying the vector field of a system from snapshot pairs, including data sampled from different trajectories.
Source: A. Problem statement · p. 2Method
The method lifts data into observable space to identify a finite-dimensional Koopman operator linearly, then derives a vector-field fit from the projected operator.
Source: B. Lifting technique using the Koopman… · p. 2Results
The authors report that the method is robust to noise and efficiently recovers vector fields for different system behaviors, including unstable and chaotic systems.
Source: §I · p. 1, §VI · p. 6Takeaways & Limitations
For network reconstruction, the method identifies links and provides their nature and weight.
Source: C. Network reconstruction · p. 6The stated system class is limited to polynomial vector fields.
Source: §I · p. 1
Abstract
from arXiv · showhide
We exploit the key idea that nonlinear system identification is equivalent to linear identification of the socalled Koopman operator. Instead of considering nonlinear system identification in the state space, we obtain a novel linear identification technique by recasting the problem in the infinite-dimensional space of observables. This technique can be described in two main steps. In the first step, similar to the socalled Extended Dynamic Mode Decomposition algorithm, the data are lifted to the infinite-dimensional space and used for linear identification of the Koopman operator. In the second step, the obtained Koopman operator is "projected back" to the finite-dimensional state space, and identified to the nonlinear vector field through a linear least squares problem. The proposed technique is efficient to recover (polynomial) vector fields of different classes of systems, including unstable, chaotic, and open systems. In addition, it is robust to noise, well-suited to model low sampling rate datasets, and able to infer network topology and dynamics.
I. INTRODUCTION
The paper connects Koopman-based analysis of trajectory data with recovery of a nonlinear vector field by identifying a linear operator in observable space. Its two-step method lifts snapshot data to identify the operator, then uses it to recover the vector field.
I. INTRODUCTION
- The proposed method bridges analysis of trajectory data and vector-field descriptions by identifying the Koopman operator from data and connecting it back to the nonlinear system.The paper frames this as connecting two directions of analysis.Source: §I · p. 1
- The approach relies on linear techniques and avoids estimating time derivatives, making it suitable for low-sampling data and robust to measurement and process noise.The stated scope is polynomial vector fields, including unstable and chaotic systems.Source: §I · p. 1
A. Problem statement
- The identification target is the vector field F, given snapshot pairs sampled at a common period; points may come from different trajectories.Measurement noise is modeled as Gaussian with zero mean and standard deviation σ_meas.Source: A. Problem statement · p. 2
B. Lifting technique using the Koopman operator
- The method lifts data into observable space, identifies a finite-dimensional Koopman operator linearly, then derives the vector field from the projected operator.The first step corresponds to a component of EDMD [12]; the second uses an analytic derivation of the infinitesimal generator and linear least squares.Source: §I · p. 1, B. Lifting technique using the Koopman… · p. 2, Figure 1 · p. 2
III. NUMERICAL METHOD
The numerical method makes the Koopman-based procedure tractable by working in a finite-dimensional function subspace. It estimates an operator matrix from data and connects that representation to the nonlinear vector field.
III. NUMERICAL METHOD
- The infinite-dimensional observable-space method is implemented in a finite-dimensional subspace spanned by N linearly independent functions.The stated basis uses monomials with total degree at most m; other bases, including Fourier and radial basis functions, could also be considered.Source: §III · p. 2
- Projecting the Koopman operator onto the chosen subspace yields an approximate finite-dimensional linear representation that is valid globally in state space, not a local linearization.The method computes the operator from data and then connects its matrix to the nonlinear system's vector field.Source: §III · pp. 2–3
A. Step 1: Lifting of the data and Koopman operator identification
The method lifts snapshot pairs into a finite basis of observables and estimates a Koopman matrix by least squares. It then uses that representation to identify the coefficients of a polynomial vector field.
A. Step 1: Lifting of the data and Koopman operator identification
- Snapshot pairs are lifted to basis-function values, and the resulting observable data provide snapshots for estimating the Koopman operator.The lifting is described as an EDMD component.Source: A. Step 1: Lifting of the data and… · p. 3
- A finite-dimensional Galerkin projection yields the Koopman matrix as a least-squares fit to the lifted data.The matrix represents the projection of the Koopman operator over the sampling period.Source: A. Step 1: Lifting of the data and… · p. 3
B. Step 2: Identification of the vector field
- Assuming a polynomial vector field of total degree at most mF reduces its identification to estimating nNF coefficients.NF counts the monomials of total degree at most mF.Source: B. Step 2: Identification of the… · p. 3
1) Derivation of the Koopman infinitesimal generator:
The vector-field coefficients determine the projected Koopman infinitesimal generator on a monomial basis. The derivation expresses that projection through finite matrices representing the generator’s action on monomials.
1) Derivation of the Koopman infinitesimal generator:
- Each unknown vector-field coefficient contributes to the projection of the infinitesimal generator onto the monomial basis.The method derives the matrix as a function of the unknown coefficients.Source: 1) Derivation of the Koopman… · p. 3, 2) Computation of the vector field · p. 3
- The operators for individual vector-field coefficients map degree-at-most-m1 polynomials to degree-at-most-m2 polynomials, where m2 = m1 + mF − 1.Their monomial-basis representations are finite matrices with N1 ≤ N2.Source: 1) Derivation of the Koopman… · p. 3
- The monomial-basis matrix entries encode the action of each coefficient operator using the exponent index function Ψ.The entry is ψj(l) when Ψ(i) = Ψ(k) + Ψ(l) − ej, and zero otherwise.Source: 1) Derivation of the Koopman… · p. 3
2) Computation of the vector field:
The vector-field coefficients are estimated from a linear, overdetermined system based on the projected generator. The procedure has sampling and matrix-logarithm constraints, while alternative semigroup fitting is described as nonlinear least squares.
2) Computation of the vector field:
- With too few data points, the estimated Koopman matrix can have negative real eigenvalues, preventing a real matrix logarithm.The principal logarithm is real only when the matrix’s real eigenvalues are not negative.Source: 2) Computation of the vector field · p. 4
- When the generator has complex eigenvalues, the sampling period must be sufficiently small to keep the scaled eigenvalues within the specified strip.The stated strip is |ℑ{z}| < π; violating the condition alters accuracy.Source: 2) Computation of the vector field · p. 4
- The coefficient equations are overdetermined, and their least-squares solution approximates the vector-field coefficients.The construction restricts the data matrix to monomials whose generator images lie in the basis span, where the finite Galerkin projection is exact.Source: 2) Computation of the vector field · p. 4
- For m1 = 1, the effective equality count equals nNF, so the coefficient problem is not overdetermined and can be solved directly..Source: 2) Computation of the vector field · p. 4
- Identifying the Koopman semigroup instead leads to a nonconvex nonlinear least-squares problem; simulations suggest solving (17) gives better results.The authors leave LASSO-style sparsity promotion for future research.Source: 2) Computation of the vector field · p. 4
IV. EXTENSIONS TO OPEN SYSTEMS
The method is presented as suitable for open systems driven by known inputs or white process noise.
IV. EXTENSIONS TO OPEN SYSTEMS
- The proposed method is described as well-suited to open systems driven by a known input or white process noise..Source: §IV · p. 4
A. Systems with inputs
For systems with inputs, the method applies the Koopman lifting to an augmented state that includes a constant input. Its use between snapshots relies on a zero-order hold assumption, or equivalently a sufficiently high sampling rate.
A. Systems with inputs
- The flow is defined by the solution from the initial state under the specified input, and observables are functions of state and input.The input signal maps from R+ to the input space.Source: A. Systems with inputs · p. 4
- Under a zero-order hold, the input is treated as an additional state variable that remains constant, yielding the classical Koopman operator for the augmented system.The augmented state is (x, u), with u̇ = 0.Source: A. Systems with inputs · p. 4
- Identification uses snapshot pairs ([x_k, u_k], [y_k, u_k]) and otherwise follows the procedure for the augmented state space R^(n+p).The method applies when the input is constant between snapshots, or sampling is sufficiently fast.Source: A. Systems with inputs · p. 4
B. Process noise
The method extends to stochastic systems with process noise without changing its main identification scheme. A projected Laplacian accounts for diffusion, while the coefficient equalities used in the second step remain unchanged.
B. Process noise
- The stochastic formulation models process noise as white noise and defines the associated Koopman semigroup and infinitesimal generator.The generator includes a Laplacian that accounts for diffusion.Source: B. Process noise · p. 4
- The numerical scheme does not need adaptation for process noise: the first step still identifies U, while the second includes the projected Laplacian matrix.The Laplacian accounts for the diffusion term.Source: B. Process noise · pp. 4–5
- Diffusion leaves the coefficient equalities unchanged, so their solution remains the same as in equation (18).The diffusion term modifies entries that are zero in the no-process-noise case.Source: B. Process noise · p. 5
V. NUMERICAL EXAMPLES
The numerical examples test recovery across periodic, unstable, and chaotic systems, with the method performing well overall. Accuracy is somewhat lower for unstable and chaotic cases and improves with more data points.
V. NUMERICAL EXAMPLES
- The examples demonstrate recovery of vector fields for unstable, chaotic, and open systems, and also apply the method to network reconstruction.The section also reports a test on periodic, unstable, and chaotic systems.Source: §V · p. 5
A. Periodic, unstable, and chaotic systems
- Accuracy increases with the number of data points, which is low in these examples; a comprehensive study of this effect is outside the paper’s scope.The examples use measurement noise with standard deviation 0.01.Source: §V · p. 5
- The method performs well across periodic, unstable, and chaotic systems, although NRMSE is slightly larger for the unstable and chaotic cases.The reported RMSE and NRMSE are averages over 10 experiments.Source: Table I · p. 5, A. Periodic, unstable, and chaotic… · p. 5
B. Input and process noise
The input and process-noise examples show vector-field estimation under forcing and strong noise. The network experiment reports accurate coefficient and link recovery, including link type and weight, with accuracy declining as the dataset shrinks.
B. Input and process noise
- The identification method estimates the vector field, including the forcing term, with an RMSE of 0.025 over all coefficients, averaged over 10 simulations, using input u(t) = cos(t).Source: B. Input and process noise · p. 5
- With process noise σ_proc = 1 and measurement noise σ_meas = 0.01, the method recovers the vector field with an average RMSE of 0.078 over 10 simulations.The process noise replaces the forcing term, and trajectories are computed with the Euler-Maruyama scheme.Source: B. Input and process noise · p. 5
C. Network reconstruction
- With 500 trajectories and 1,500 data points, the 12-dimensional network experiment estimates 5,460 coefficients with RMSE = 0.021.The data comprise three snapshots per trajectory at times 0, 0.5, and 1.Source: C. Network reconstruction · p. 6
- As the number of data points decreases, results become less accurate but remain acceptable: RMSE is 0.106 with 400 trajectories and 0.151 with 300; sparsity-promoting methods such as LASSO could further improve accuracy.Source: C. Network reconstruction · p. 6
- At a threshold of 0.1, network reconstruction achieves a true positive rate of 0.875 and a false positive rate of 0.023, while also identifying link type and weight.A link is identified when at least one coefficient associated with the corresponding state interaction exceeds the threshold.Source: C. Network reconstruction · p. 6
VI. CONCLUSION
The proposed lifting method recasts nonlinear system identification as linear Koopman-operator identification and is reported to suit low-sampling-rate data, tolerate noise, and recover vector fields efficiently. The paper identifies non-polynomial recovery with other bases and convergence of the finite-dimensional truncation as future work.
VI. CONCLUSION
- The lifting method recasts nonlinear system identification as linear Koopman-operator identification and is reported to suit low-sampling-rate data, tolerate noise, and recover vector fields efficiently.The method uses linear techniques and does not require estimating time derivatives.Source: §VI · p. 6
- Figure 2 reports recovery of all coefficients of a 12-dimensional polynomial vector field, distinguishing exact nonzero values from estimates by color.Exact nonzero values are blue; estimated values are red.Source: Figure 2 · p. 6, §VI · p. 6
- The paper leaves recovery of non-polynomial vector fields using other bases and theoretical convergence of the finite-dimensional truncation for future work.The authors characterize the results as preliminary.Source: §VI · p. 6