Kernel-Adaptive Physics-Informed Shallow Meta-Learning for Parametric Linear PDEs

Vikas Dwivedi, Monica Sigovan, Sixou Bruno

Keywords: Physics-informed neural networks, Meta-learning, Parametric PDEs, Radial basis functions

TL;DR: We replace weight-based meta-learning for PDEs with kernel-geometry adaptation, enabling interpretable, physics-informed meta-solvers for parametric linear PDEs.

Abstract: Parametric families of linear partial differential equations (PDEs) arise widely in scientific and engineering applications, where solutions depend continuously on physical parameters such as source location, transport velocity, or diffusion strength. Many physics-informed and neural operator approaches either require retraining for each parameter instance or rely on high-dimensional learned representations that can be difficult to interpret and control. We propose a kernel-adaptive physics-informed meta-learning framework for parametric linear PDEs in which task variation is modeled through low-dimensional deformations of kernel distributions rather than task-specific network weights. The solution is represented as a linear combination of physics-aligned kernel functions whose centers, widths, sparsity gates, and optionally coefficients are conditioned on PDE parameters through a lightweight meta-parameterization. By exploiting linearity, the formulation preserves superposition and can be trained without inner-loop optimization; in the studied settings, it admits an interpretation in terms of parameter-dependent kernel responses analogous to Green's-function behavior. We evaluate the method on representative elliptic (Poisson), hyperbolic (advection), and mixed parabolic--hyperbolic (advection--diffusion) PDE families and compare against parameter-conditioned neural baselines including PI-DeepONet and FiLM--HyperPINN. Across these cases, the proposed approach achieves $\mathcal{O}(10^{-2})$--$\mathcal{O}(10^{-1})$ relative $L^2$ errors within the training parameter ranges without per-instance retraining, and exhibits structured error growth under extrapolation consistent with resolution and parameter-support limits of the learned kernel manifold. These comparisons illustrate how separable operator decompositions, feature-wise modulation, and adaptive kernel geometry induce distinct inductive biases under parameter shift.

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Journal Notes: We are very interested and committed to extend our work for a journal paper. The mathematical framework, novelty, interpretability and the proof of concept examples are already provided. If given an opportunity, we would provide the detailed comparison of proposed method with neural operators on all the benchmark cases, and release the full codes for reproducibility. We expect to finish the paper by the end of this month, i.e., March 2026.

Submission Number: 62