Amortized Physics-Informed Learning via Generative Initialization of Radial Basis Functions

Physics-Informed Neural Networks (PINNs) commonly optimize a new solution model for each PDE instance, making repeated changes in parameters, boundary conditions, or forcing terms costly. GI-RBF addresses this limitation through a decoupled offline–online framework combining an explicit anisotropic Radial Basis Function (RBF) solver with a task-conditioned generative initializer. The initializer amortizes the search for the complete RBF state—including coefficients, centers, and kernel geometry—evaluated once per query; its output is then refined directly under the deployment physics objective. Analytic derivatives and the localized explicit representation enable inexpensive physics updates, while learned initialization reduces the refinement needed to reach useful accuracy. Experiments on four canonical PDE families and a transient $(2+1)$-dimensional Flow Mixing problem show improved finite-budget adaptation over standard RBF initialization across interpolation and extrapolation regimes. The resulting inference-through-adaptation paradigm reaches an accurate solution through fast, physics-based refinement at deployment.