Fisher-Informed Training of Neural Operators for Reliable PDE Inversion

Neural operators have shown strong performance for PDE-solution learning; however, their effectiveness in PDE-constrained optimization, such as inversion, remains limited, in part because inaccurately learned derivatives can mislead gradient-based updates. Standard training typically optimizes only the forward least-squares fit, so during iterative inversion the optimization variable can drift outside the training distribution, further degrading quality of derivative information and inversion performance. To address this limitation, we propose a training algorithm designed for inversion: the surrogate is trained not only to predict forward outputs but also to learn observation-relevant gradients, which leads to Fisher-information directed parameter updates. We evaluate forward and gradient accuracy, as well as the inversion trajectory, on problems governed by PDEs—Darcy flow and laminar (incompressible) Navier–Stokes—which are relevant to porous-media fluid flow and energy-storage applications. Our results suggest that training for gradient fidelity with forward accuracy unlocks a pathway to reliable, efficient, neural operator–based inversion.