Neural Operators

Neural operators such as FNO and DeepONet learn mappings between infinite-dimensional function spaces, achieving resolution invariance and orders-of-magnitude speedups over classical solvers. FNO achieves competitive mean accuracy on stochastic Burgers but has no variance head and cannot provide stochastic residual information. FNO outperforms MNO on Gray-Scott because it can allocate its full capacity to mean prediction without a competing variance head. Training standard neural operators with L2 loss on stochastic PDE data forces convergence to the conditional mean, discarding the full terminal law including spatial variance and tail structure.