Basis-Adaptive Learned Asymptotic Nonlocal Inversion Neural Operator (BALANI-NO) for PDE Surrogates and Gravitational Lensing

Spectral neural operators parameterize each layer by a learned Fourier multiplier, or symbol, which is usually a lookup table over the lowest modes. For many elliptic operators the symbol is instead a rational function of frequency. We introduce BALANI-NO, a neural operator whose symbol is the sum of a learned pole (rational) branch in the isotropic magnitude $|k|$ and a small low-mode table. The pole branch specifies the symbol at all frequencies from a few coefficients, favors no coordinate axis, and does not depend on the grid. We compare BALANI-NO with ten baselines at matched parameter budgets, with seven seeds each, on Darcy flow, Navier–Stokes vorticity, and gravitational-lensing deflection fields, where models trained on warm dark matter are tested on cold and axion dark matter. BALANI-NO has the lowest zero-shot error on Darcy ($0.1194$ vs. $0.1300$ for FNO), on Navier–Stokes, and on every zero-shot lensing split. It also has the lowest in-distribution error on Navier–Stokes and subhalo lensing, and ties with U-FNO on full-field lensing. Two smaller-scale additional experiments, on real (non-synthetic) cosmological density maps and real geomagnetic survey data, support the same pattern of best or near-best zero-shot transfer.