Modal-Centric Field Inversion via Differentiable Proper Orthogonal Decomposition

TL;DR

This work addresses the prohibitive cost and ill-conditioning of inverse problems involving high-dimensional spatio-temporal fields by reformulating calibration in the reduced space of POD modes. It introduces Modal-Centric Field Inversion (MCFI) with a differentiable POD adjoint framework that yields gradient-based optimization in modal space, requiring only a single unsteady adjoint solve regardless of the number of modes. The approach is demonstrated on one- and two-dimensional modified Burgers equations, showing accurate adjoint gradients (validated against finite differences) and successful mode-matching optimizations, with multi-mode objectives demonstrating robustness and interpretability. The results indicate that MCFI can regularize inverse problems, dramatically reduce computational burden, and enable scalable inverse design and model calibration for unsteady, high-dimensional systems, with extension potential to CFD and turbulence-model calibration.

Abstract

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.

Figures (12)

• Figure 1: Schematic of MCFI. FI (left) matches point-wise values in the full spatio-temporal space, leading to high-dimensional and sometimes ill-conditioned optimization. MCFI (right) reformulates the problem in the reduced POD state space, matching dominant coherent structures instead of individual field values. The differentiable POD (\ref{['eq:base_adj_expand']}) provides the adjoint-based gradients that enable efficient optimization in the modal space.
• Figure 2: Selected snapshots of the 1D viscous Burgers solution for the uniform multiplier case ($\alpha(x)=0.1$), with the legend denoting $t=\{0.00,0.80,1.60,2.40\}$ . The smooth upwind/central discretization captures the pulse interaction and subsequent viscous decay resolved in the POD snapshots.
• Figure 3: Comparison of the initial, optimized, and target Gaussian control profiles for the 1D mode-matching problem. The optimized profile recovers the target everywhere while preserving the smoothing enforced by the Gaussian basis.
• Figure 4: First POD mode associated with the initial control, the optimized solution, and the target trajectory. The optimized mode collapses onto the target mode, verifying that the POD-adjoint coupling transports the sensitivities without loss of accuracy.
• Figure 5: Semilog plot of the IPOPT objective history for the objective function $f_1$. The adjoint-driven iterations reduce the mode loss by more than eleven orders of magnitude before reaching the prescribed iteration cap.