reducedorder

Reduced-order modeling (ROM), also called model order reduction (MOR), is the development of simplified mathematical models that reproduce the essential input-output behavior of a high-fidelity system using far fewer degrees of freedom. ROMs speed up simulation, enable real-time control, and reduce computational cost while aiming to preserve accuracy.

Most approaches are projection-based. A high-dimensional state x is approximated as x ≈ V x_r, with V containing basis vectors. Substituting into the original model and projecting yields a reduced system with matrices Ar, Br, Cr, Dr. Common techniques include balanced truncation (Gramians to identify removable states), proper orthogonal decomposition (POD) from data, and Krylov or moment-matching methods. Data-driven and nonlinear MOR methods also exist.

For linear time-invariant systems, the reduced model has ẋ_r = Ar x_r + Br u and y = Cr x_r + Dr u. In nonlinear problems, reduced models may use empirical gramians, operator inference, or other nonlinear reduction strategies. Validation against the full model is essential to ensure fidelity, stability, and robustness within the intended regime.

ROMs are widely used in aerospace, automotive, energy, fluids, electronics, and mechanical engineering for design optimization, parametric studies, and real-time simulation. Their effectiveness depends on selecting an appropriate basis, preserving important dynamics, and rigorous verification across anticipated inputs and parameters.