# Fourier Neural Operator (FNO)

Training five operator models on the one-dimensional discrete Poisson problem shows that error is not learned wavenumber by wavenumber but is retained along small-eigenvalue directions of the kernel that governs the training update. Sine-aligned spectral and linear Fourier neural operators give each wavenumber its own rate, so error collects at high wavenumbers; dense networks have eigendirections close to random in the sine basis, so error is nearly equal at every wavenumber and relative error grows only because the true solution spectrum decays. Removing the minibatch cross terms leaves the GELU dense network with under 0.1% of squared error below ηλ≈8×10⁻⁵ under either forcing.

Training five operator models on the one-dimensional discrete Poisson problem shows that error is not learned wavenumber by wavenumber but is retained along small-eigenvalue directions of the kernel that governs the training update. Sine-aligned spectral and linear Fourier neural operators give each wavenumber its own rate, so error collects at high wavenumbers; dense networks have eigendirections close to random in the sine basis, so error is nearly equal at every wavenumber and relative error grows only because the true solution spectrum decays. Removing the minibatch cross terms leaves the GELU dense network with under 0.1% of squared error below ηλ≈8×10⁻⁵ under either forcing.

Training neural operators needs expensive numerical solution labels, while PDE inputs and governing equations are cheap physical information. PI-Init spends that information on initialization: it pretrains on unlabeled inputs with physics losses, then keeps the parameters for supervised fine-tuning. At matched label budgets, FNO prediction error falls 53.3% on Burgers and 62.7% on 1D Advection, and on Burgers 200 labeled trajectories beat supervised-only training with 2,000.

Training neural operators needs expensive numerical solution labels, while PDE inputs and governing equations are cheap physical information. PI-Init spends that information on initialization: it pretrains on unlabeled inputs with physics losses, then keeps the parameters for supervised fine-tuning. At matched label budgets, FNO prediction error falls 53.3% on Burgers and 62.7% on 1D Advection, and on Burgers 200 labeled trajectories beat supervised-only training with 2,000.