Meta-SPINN: meta-learned basis adaptation for parametric PDEs

Abstract

Physics-informed machine learning of parametric partial differential equation (PDE) families enables rapid prediction across varying physical conditions, yet the resulting task representations are commonly embedded in latent neural features that are difficult to interpret physically. This raises the question of whether a parametric neural PDE solver can make explicit how physical task parameters reorganize the underlying solution representation. To address this gap, we introduce Meta-Sparse, Physics-based, and partially Interpretable Neural Network (SPINN), which maps task parameters to a shallow RBF model with inspectable coefficients, centers, scales, and directional parameters. We show that, across elliptic, transport, advection–diffusion, variable-coefficient, and nonlinear PDE families, the learned bases adapt to and organize around the dominant physical solution structures, including localized forcing responses, characteristic-aligned transport trajectories, diffusion-broadened space–time corridors, and viscous shock fronts. Meta-SPINN works both as a direct predictor for unseen tasks and as a task-aware initializer for subsequent single-instance residual-guided refinement, providing reusable predictions together with an interpretable visualization of how solution geometry changes across a parameter family.