Meta-Learned Basis Adaptation for Parametric Linear PDEs

2026-04-10

AI Summary

This work addresses the challenge of simultaneously achieving generalization, accuracy, and efficiency in solving parametric families of linear partial differential equations (PDEs) by proposing the KAPI framework. KAPI leverages meta-learning to construct task-adaptive Gaussian basis functions with tailored geometric structures and integrates a physics-informed extreme learning machine (PIELM) for a one-shot least-squares correction. The method introduces, for the first time, an interpretable mechanism for adaptive basis construction by explicitly modeling the mapping from PDE parameters to basis function geometry, synergistically enhancing solution accuracy through physical constraints. Experiments across four classes of PDE families—including diffusion and transport—demonstrate that KAPI captures physically aligned local basis structures, yielding post-correction accuracy improvements of one to two orders of magnitude over parametric PINNs and DeepONet.

Technology Category

Machine Learning: Kernel MethodsSearch and Optimization: Learning to SearchConstraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applicationsGraph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphsUser Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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.

Problem

• parametric PDEs
• basis adaptation
• physics-informed learning
• meta-learning
• linear partial differential equations

Innovation

• meta-learning
• physics-informed
• adaptive basis
• parametric PDEs
• Extreme Learning Machine