A Backward Kolmogorov Neural Operator for Stochastic Partial Differential Equation

Solutions of a stochastic partial differential equation can follow different trajectories from the same initial state. We propose the Backward Kolmogorov Neural Operator (BKNO) for predicting their conditional mean when the driving noise is not observed. The method extends Koopman-based neural approximation of deterministic equations to equations with process noise. A nonlinear map represents the field by spatial observables. Their low-frequency Fourier coefficients are advanced by a matrix constructed from a generator estimate and the sampling interval. A local convolution accounts for spatial variation omitted by truncation, and a projection returns the predicted physical field. Stochastic Koopman theory describes the conditional evolution used in the spectral step. The complete approximation is a neural operator from fields to fields. When the noise vanishes, the underlying evolution reduces to the deterministic setting of the Koopman neural operator (KNO). Numerical experiments on stochastic phase-field equations, neural-field equations, and the Kuramoto--Sivashinsky equation show improved predictions of fields and physical quantities. Validation experiments on speech electrocorticography recordings also show improved forecasts of neural activity and predictions of speech-related responses.