Spherical Laplace Neural Operators with Spatio-Temporal Modeling

Neural operators provide efficient surrogate models for dynamical partial differential equations (PDEs). Existing spherical neural operators have advanced the modeling of global spatial interactions through spherical spectral representations, while transient temporal dynamics remain implicit in learned discrete-time mappings. To address this limitation, we introduce the Spherical Laplace Neural Operator (SLNO), which combines spherical Fourier modeling in space with Laplace-domain modeling in time. By conditioning temporal transfer functions on the Laplace–Beltrami spectrum, SLNO explicitly represents scale-dependent decay and oscillatory dynamics through learnable poles and channel-valued residues. We further derive analytic initial-state and trajectory-driven responses to construct and propagate internal temporal features within nonlinear neural operator blocks. Extensive Experiments on controlled linear fast–slow system, spherical PDEs and global weather forecasting demonstrate the accurate and stable long-horizon prediction of SLNO. These results demonstrate the potential of explicit spatio-temporal spectral modeling for learning long-term spherical dynamics.