Fisher-Informed Neural Operators for PDE-Constrained Inversion

Scientific inverse problems often require repeated evaluations of high-dimensional PDE forward models, motivating neural surrogates as efficient alternatives to accurate numerical solvers. However, even when a surrogate closely approximates a numerical solver in $L^2$, using it as the forward model can yield highly inaccurate parameter estimates. Here, we study gradient-based minimization of observation error for parameter inversion and propose training neural surrogates using the leading eigenvectors of an amortized Fisher information matrix. The resulting Fisher-informed Neural Operator (FINO) is a derivative-informed neural operator that learns the forward model Jacobian only along directions relevant to maximum-likelihood optimization. For computational efficiency, FINO uses randomized SVD to obtain an amortized set of Fisher-information eigenvectors. Our theoretical analysis relates Fisher directions to maximum-likelihood estimate recovery and characterizes how the effective rank of the inverse-problem Jacobian enters through spectral truncation and conditioning. We evaluate FINO on three forward problems with widely varying sensitivity structure: Darcy flow, two-dimensional Navier–Stokes flow, and seismic imaging with the acoustic wave equation. On Navier–Stokes, FINO reduces mean parameter RMSE by $42.8$% and $25.1$% relative to FNO trained only with $L^2$ and DINO, respectively. Controlled spectral-decay experiments show how FINO’s benefit depends on the Fisher spectrum. Its amortizable offline cost also makes FINO practical for scientific inverse problems.