Analytical Neural Operator (ANNO): Learning Reusable Analytical Structures for Time Evolution in PDE Families

Neural operators have emerged as an efficient framework for learning solution operators of PDE families. For time-dependent PDEs, existing learning-based solvers typically learn temporal responses from transient trajectories, either by predicting full spatiotemporal solution fields or by learning recursive propagators. For systems governed by known PDEs, however, the temporal evolution is already constrained by the governing equations. We propose the Analytical Neural Operator (ANNO), which learns reusable analytical structures from steady-state problems and reuses them for time evolution within PDE families. ANNO explicitly represents the lifting function, spatial eigenmodes, and modal coefficients, and constrains the learned modes through the corresponding spatial eigenproblem. After steady-state training, the Learnable Mode Network and Learnable Coefficient Network are frozen and reused for time-dependent PDEs in the same family, while the dynamic modal coefficients are advanced solely according to the target PDE dynamics. This requires neither transient training trajectories nor network retraining. Across five experiments, ANNO accurately recovers steady-state solutions and spatial eigenstructure, and reuses the learned analytical structure for 1D diffusion, 2D wave, and 2D engineering thermal dynamics under nonzero initial conditions, spatiotemporally varying sources, time-varying boundaries, and varying physical parameters. On these transient tasks, ANNO achieves mean relative $L_2$ errors of 0.2531%, 1.8646%, and 1.0506%, respectively, while substantially outperforming fully supervised transient baselines trained on complete trajectories. Ablation studies further support the effectiveness and computational efficiency of the proposed analytical structure. These results show that reusable analytical structures learned from steady-state problems can support accurate and data-efficient time evolution within PDE families.