Data-Driven Reduced Order Models Under Physical Constraints.

We propose a Jacobian-free projection framework for constructing physics-constrained, data-driven reduced-order models (ROMs) for discrete latent dynamical systems, such as those introduced in [1,2]. The dynamics are represented by the quadratic discrete-time evolution law. Unconstrained black-box models can fit data accurately, yet they often suffer from instability and energy imbalance during long-term propagation. In contrast, our framework embeds physical constraints, such as energy conservation, dissipation, or dynamical stability, directly into the identification process, yielding ROMs that remain faithful to the governing physics.

Operator identification is carried out with a Jacobian-free conjugate-gradient algorithm that solves the associated transpose Sylvester optimality problem entirely in a Krylov subspace. Numerical examples show that the resulting ROMs provide a robust, principled approach to modeling complex nonlinear systems while maintaining both data fidelity and adherence to essential physical laws.

References

[1] Ayoub, R. and Oulghelou, M. and Schmid, P. Improved Greedy Identification of Latent Dynamics with Application to Fluid Flows. Computer Methods in Applied Mechanics and Engineering, 2025.

[2] Oulghelou, M. and Ammar, A. and Ayoub, R. Greedy identification of latent dynamics from parametric flow data. Computer Methods in Applied Mechanics and Engineering, 2024.