cs.LGApr 9, 2026

Tensor-based computation of the Koopman generator via operator logarithm

Authors: Tatsuya Kishimoto, Jun Ohkubo

Organizations: Graduate School of Science and Engineering, Saitama University 255 Shimo-Okubo, Sakura-ku, Saitama 338–8570, Japan

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Abstract

Identifying governing equations of nonlinear dynamical systems from data is challenging. While sparse identification of nonlinear dynamics (SINDy) and its extensions are widely used for system identification, operator-logarithm approaches use the logarithm to avoid time differentiation, enabling larger sampling intervals. However, they still suffer from the curse of dimensionality. Then, we propose a data-driven method to compute the Koopman generator in a low-rank tensor train (TT) format by taking logarithms of Koopman eigenvalues while preserving the TT format. Experiments on 4-dimensional Lotka-Volterra and 10-dimensional Lorenz-96 systems show accurate recovery of vector field coefficients and scalability to higher-dimensional systems.

Figures & tables

Figure 1: Segment of the vdP oscillator trajectory data used for EDMD. The trajectory is generated from the initial condition [x1,x2]=[1.0,1.0][x_{1},x_{2}]=[1.0,1.0] , and this plot shows the part from t=0t=0 to t=10t=10 sampled with a sampling interval of Ts=0.1T_{\mathrm{s}}=0.1 .
Figure 2: Illustration of the proposed method for computing the Koopman generator in the TT format. (a) shows the conventional matrix-logarithm-based approach, whereas (b) summarizes the proposed approach based on the eigendecomposition relationship. In the proposed method, the logarithm is applied to the eigenvalues rather than to the Koopman matrix, enabling efficient computation in high-dimensional settings. The Koopman matrix and its eigendecomposition are shown in parentheses to indicate that the eigenvalues and eigentensors can be computed directly using AMUSEt.
Figure 3: Tensor contraction process for computing the Koopman generator in the TT format. Ξ\mathbf{\Xi} is the Koopman eigentensor, Σ\Sigma is the diagonal matrix of the eigenvalues of the Koopman generator, and Ξ+\mathbf{\Xi}^{+} is the pseudo-inverse of the Koopman eigentensor. By contracting the last mode of Σ\mathbf{\Sigma} and the first mode of Ξ+\mathbf{\Xi}^{+} , we can obtain the Koopman generator L\mathbf{L} in the TT format.
 Ξ+={UX,rΣX,r−1{A+}T}T.\mathbf{\Xi}^{+}=\left\{\mathbf{U}_{X,r}\Sigma^{-1}_{X,r}\left\{A^{+}\right\}^{\mathsf{T}}\right\}^{\mathsf{T}}. 
Algorithm 1 Proposed method for Koopman generator identification in the TT format
Figure 4: Vector field coefficients obtained for the Lotka–Volterra system. The horizontal axis represents the indices of the coefficient vector, and the vertical axis represents the values of the coefficient vector. The blue dots represent the true coefficients, and the orange dots represent the estimated coefficients. All coefficients are nearly identical between the true and estimated values.
TermAbsolute error
x2x10x_{2}x_{10}3.67×10−83.67\times 10^{-8}
x3x1x_{3}x_{1}5.18×10−65.18\times 10^{-6}
x4x2x_{4}x_{2}4.69×10−84.69\times 10^{-8}
x5x3x_{5}x_{3}6.75×10−76.75\times 10^{-7}
x6x4x_{6}x_{4}1.56×10−71.56\times 10^{-7}
x7x5x_{7}x_{5}5.78×10−85.78\times 10^{-8}
Table 1: Absolute errors in the estimated coefficients of the xi+1xi−1x_{i+1}x_{i-1} terms in the Lorenz–96 system. The absolute error for each coefficient is defined as the absolute value of the difference between the true and estimated values, i.e., ∣w^k,i−1∣|\hat{w}_{k,i}-1| .
TermAbsolute error
x1x2x_{1}x_{2}9.46×10−79.46\times 10^{-7}
x2x3x_{2}x_{3}4.22×10−64.22\times 10^{-6}
x3x4x_{3}x_{4}1.62×10−61.62\times 10^{-6}
x4x5x_{4}x_{5}9.29×10−79.29\times 10^{-7}
x5x6x_{5}x_{6}1.19×10−71.19\times 10^{-7}
x6x7x_{6}x_{7}4.37×10−74.37\times 10^{-7}
Table 2: Absolute errors in the estimated coefficients of the xixi+1x_{i}x_{i+1} terms, which are not present in the Lorenz–96 system. The absolute error for each coefficient equals the absolute value of the estimate, since the true value is zero.
Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Graphical representation of the reduced-matrix computation. The tensors UX,r\mathbf{U}_{X,r} and UY,r\mathbf{U}_{Y,r} are left-orthonormalized so that the contractions can be performed efficiently. Since we define Ψ(Y)=KΨ(X)\mathbf{\Psi}(Y)=K\mathbf{\Psi}(X) , the tensor contraction sequence differs from that in the original algorithm.

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