Beyond Fourier and Laplace: The Hankel Neural Operator for Learning Solution Operators of Differential Equations

Fourier and Laplace neural operators (FNO, LNO) learn kernel integral operators in a transform domain evaluated on a periodic grid. FNO has no explicit mechanism for boundaries, initial states or transients; LNO adds pole--residue symbols but evaluates them on the discrete Fourier grid with unconstrained poles. FNO's error is large on undamped oscillators and grows under grid refinement. We introduce the Hankel neural operator (HNO), derived from the structure of the solution operator. We prove that the solution operator of a well-posed linear constant-coefficient boundary-value problem on an interval is a Toeplitz operator with a rational kernel plus a finite-rank Hankel-type operator carrying boundary information. Classical results of Kronecker, Nehari and Adamyan--Arov--Krein characterise this kernel class: stable poles, finite Hankel rank and norm bounds independent of the interval length. HNO realises the class in three ways: a non-periodic convolution evaluated by the FFT, with a causal form; a closed-loop Lur'e operator for dynamics without fading memory; and a rational Fourier multiplier that contains FNO as a special case. Trained from data alone, HNO has the lowest mean test error on all nine LNO benchmarks and on the Burgers, Darcy and Navier--Stokes benchmarks of FNO; two of these differences, and the Navier--Stokes comparison with a parameter-matched FNO, are not statistically significant. On the six ODE problems it lowers the better baseline's mean error 2.1--5.2 times, and its Darcy error stays nearly constant from 85² to 421² grids, where FNO's grows eightfold. Ablations support each component. Code and data: {}.