Learning Dynamics from Data by Future-Informed Regression of Evolution

The data-driven modeling of nonlinear dynamical systems using the Koopman operator has become a widely adopted framework for spectral analysis, prediction, and control. However, classical Koopman-based methods are typically restricted to observables defined on the system state at a single time instant, which limits their expressivity for systems exhibiting temporal correlations, memory effects, or multi-step interactions. In this work, we introduce a generalized linear mapping operator designed to establish the optimal linear relationship between two complex, trajectory-dependent observables defined over an extended state space that incorporates both past and future dynamics. By allowing heterogeneous input–output observable spaces, the proposed framework systematically captures temporal dependencies, coupled dynamics, and physically informed features, extending the applicability of Koopman-based data-driven modeling. Numerical experiments on benchmark systems, including the SIR epidemic model, a two-mass spring–damper system, and a forced harmonic oscillator, demonstrate improved reconstruction accuracy and spectral representation compared to standard approaches. In particular, the proposed method achieves relative reconstruction errors as low as 5.89×10−5 for the SIR model and 8.96×10−4 for the forced harmonic oscillator, representing improvements of several orders of magnitude over classical DMD and EDMD variants. These results confirm the robustness of the new framework in capturing complex nonlinear and transient dynamics.