KAPI-LM: kernel-adaptive physics-informed learning machines for PDEs.

Physics-informed neural networks (PINNs) have emerged as mesh-free approaches for solving partial differential equations, but their learned representations are often difficult to interpret. RBF-kernel-based formulations such as sparse, physics-based, and interpretable neural networks (SPINNs) make the geometry of the physical kernels explicit and show that kernels adapt coherently around dominant solution features, motivating the question of whether all kernel geometries need to remain independent. To investigate this question, we introduce kernel-adaptive physics-informed learning machines (KAPI-LMs), a general framework for solving PDEs with structured RBF representations in which a large physical kernel dictionary is organized through a smaller set of geometric variables while retaining all physical kernels and independent expansion coefficients. The resulting hypothesis class provides explicit control over the degree of geometric sharing and can be combined with different physics-informed learning strategies. To demonstrate this flexibility, we develop compressed SPINN (cSPINN) using joint geometry-coefficient optimization, compressed physics-informed extreme learning machine (cPIELM) using variable projection for coefficient-linear PDEs, and Meta-cSPINN using task-conditioned learning for parametric PDEs. We establish approximation guarantees and quantify the effect of geometric sharing. Numerical studies across linear, nonlinear, parametric, coupled, and multiscale PDEs demonstrate substantial geometric compression at fixed physical basis resolution and reveal problem-dependent limits to sharing. They also show that different optimization strategies can discover markedly different geometries within the same hypothesis class. For parametric PDEs, Meta-cSPINN supports one-shot prediction and suggests additional organization across solution families, while PDE residuals identify where geometric sharing should be released or local resolution added. KAPI-LM thus provides a physics-informed PDE-solving framework in which geometric complexity is explicit and controllable, allowing the solution representation to be interpreted, compressed, and adaptively refined according to the physics.