Pseudo-DeepONet: Explicit Neural Operators for PDE Dynamics

Neural operators for time-dependent partial differential equations are built in both implicit and explicit forms, yet rigorous guarantees that favor one design over the other for broad classes of problems have remained scarce. We introduce the Pseudo-DeepONet (PDON),, an explicit architecture that separates spatial representation from temporal evolution in latent space, and we establish an error analysis showing that this split design can achieve strictly smaller approximation error than standard implicit operator formulations under mild regularity and spectral-decay conditions typical of many commonly studied dissipative and transport-dominated systems. Empirically, a parity-controlled inductive-bias study shows that the factorisation encourages learning the underlying dynamics rather than mere snapshot reconstruction when encoder, decoder, and capacity are matched. Among recurrent state-space temporal backbones, a linear oscillatory core (LinOSS) performs best in our suite, and we supply a theoretical error-bound analysis explaining how its second-order oscillatory structure matches temporal structure in PDE trajectories and lowers approximation error versus alternative cores. We further demonstrate resolution-invariant evaluation via band-limited spectral transfer between discretisations. Across seven widely used PDE benchmarks spanning one-, two-, and three-dimensional spatiotemporal dynamics, PDON mostly improves strong operator-learning baselines. Finally, we report results on the large-scale reanalysis dataset ERA5, showing that the same explicit factorisation carries to complex real-world spatiotemporal fields.