Meta-Learned Basis Adaptation for Parametric Linear PDEs

TL;DR

This paper introduces a hybrid physics-informed meta-learning framework, KAPI, that adaptively constructs basis functions for efficiently solving parametric linear PDEs with high accuracy.

Contribution

It presents a novel predictor-corrector approach that learns task-specific basis geometries for parametric PDEs, improving solution accuracy and interpretability.

Findings

The predictor captures physics-aligned basis placement across PDE families.

The corrector enhances accuracy, often by one or more orders of magnitude.

Compared to PINNs and DeepONet, the method offers interpretable and efficient solutions.

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.