NOAH: Neural Operator Algebra in Hilbert Space: A New Generation of Deep Learning for PDEs Beyond Neural Operators, with Application to the Navier–Stokes Equations

We introduce Neural Operator Algebra in Hilbert Space (NOAH), a new framework that learns an adaptive family of operators instead of a single fixed mapping. Its spectral realization, F-NOAH, adds three physics-constrained generators, for advection, diffusion, and reaction, to a Fourier Neural Operator (FNO) backbone, and it restores their defining constraints exactly by closed-form projection after every optimizer step. We test the module in the regime that matters most in practice, a small training set and a short training budget: 500 initial conditions and 50 epochs on the two-dimensional Navier–Stokes equations at viscosity 10−2, trained at 64×64 and evaluated zero-shot up to 1024×1024. The baseline is the Fourier Neural Operator as published. Over fourteen independent repetitions, F-NOAH attains the lower mean squared error in all 280 paired measurements, the pooled improvement is a factor of 2.50, and every cell remains significant after Holm correction. In the best run, the error falls by 88.3% at 64×64 and by 87.2% at 1024×1024 at T=10. The NOAH module supplies a remarkable correction when data and computational resources are limited.