Modal-Centric Field Inversion via Differentiable Proper Orthogonal Decomposition

Inverse problems in computational physics often require matching high-dimensional spatio-temporal fields, leading to prohibitive computational costs and ill-conditioned optimizations. We introduce modal-centric field inversion (MCFI), a paradigm that reformulates inverse problems in the reduced space of proper orthogonal decomposition (POD) modes rather than the full physical state space. By targeting dominant flow structures instead of point-wise field values, MCFI provides a compact, physically meaningful objective that naturally regularizes the inversion and dramatically reduces computational burden. Central to this framework is the differentiable POD: an adjoint-based method that efficiently computes sensitivities of POD modes with respect to model parameters, enabling gradient-based optimization in the modal space. We demonstrate MCFI on a one and two-dimensional modified viscous Burger's equation, optimizing spatially varying coefficients to match target dynamics through mode-matching. The adjoint formulation achieves computational cost independent of parameter dimension, in contrast to finite-difference approaches that scale linearly. MCFI establishes a foundation for scalable inverse design and model calibration in unsteady, high-dimensional systems.