# Meta-Learned Basis Adaptation for Parametric Linear PDEs Source: [Meta-Learned Basis Adaptation for Parametric Linear PDEs]() Collection index: [llms.txt]() Data updated: 2026-04-14T05:00:11.146Z Authors: Vikas Dwivedi, Monica Sigovan, Bruno Sixou Published: 2026-04-10T13:00:03.000Z ## Abstract We propose a hybrid physics-informed framework for solving families of parametric linear partial differential equations (PDEs) by combining a meta-learned predictor with a least-squares corrector. The predictor, termed \\textbf{KAPI} (Kernel-Adaptive Physics-Informed meta-learner), is a shallow task-conditioned model that maps query coordinates and PDE parameters to solution values while internally generating an interpretable, task-adaptive Gaussian basis geometry. A lightweight meta-network maps PDE parameters to basis centers, widths, and activity patterns, thereby learning how the approximation space should adapt across the parametric family. This predictor-generated geometry is transferred to a second-stage corrector, which augments it with a background basis and computes the final solution through a one-shot physics-informed Extreme Learning Machine (PIELM)-style least-squares solve. We evaluate the method on four linear PDE families spanning diffusion, transport, mixed advection--diffusion, and variable-speed transport. Across these cases, the predictor captures meaningful physics through localized and transport-aligned basis placement, while the corrector further improves accuracy, often by one or more orders of magnitude. Comparisons with parametric PINNs, physics-informed DeepONet, and uniform-grid PIELM correctors highlight the value of predictor-guided basis adaptation as an interpretable and efficient strategy for parametric PDE solving. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [1D linear advection prediction on 1D linear advection in-range (test)](): 1D linear advection prediction; 1D linear advection in-range (test); Relative L2 Error: 1.166. - [2D Poisson prediction on 2D Poisson (in-range)](): 2D Poisson prediction; 2D Poisson (in-range); Relative L2 Error: 0.01195. - [1D linear advection prediction on 1D linear advection out-of-range (test)](): 1D linear advection prediction; 1D linear advection out-of-range (test); Relative L2 Error: 2.215. - [2D Poisson prediction on 2D Poisson (out-of-range)](): 2D Poisson prediction; 2D Poisson (out-of-range); Relative L2 Error: 3.569. ## Original sources and implementation [Original paper]() [PDF]()