Meta-Learning for Physics-Informed Neural Networks (PINNs): A Comprehensive Framework for Few-Shot Adaptation in Parametric PDEs

Physics-Informed Neural Networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by incorporating physical laws directly into neural network training. However, traditional PINNs require extensive retraining for each new PDE configuration, limiting their practical applicability in parametric scenarios. This work presents a comprehensive meta-learning framework for PINNs that enables rapid adaptation to new parametric PDE problems with minimal training data. We introduce four novel meta-learning architectures: MetaPINN, PhysicsInformedMetaLearner, TransferLearningPINN, and DistributedMetaPINN, each designed to address specific challenges in few-shot PDE solving. Through extensive evaluation on seven parametric PDE families including heat equations, Burgers equations, Poisson problems, Navier-Stokes equations, Gray-Scott systems, and Kuramoto-Sivashinsky equations, we demonstrate that meta-learning approaches achieve L2 relative error of 0.034 compared to 0.160 for standard PINNs, representing an 79% error reduction, while reducing adaptation time by 6.5×. Our PhysicsInformedMetaLearner consistently outperforms all baselines across 280 statistical comparisons with 92.9% significance rate. The framework includes comprehensive computational analysis showing break-even points at 13-16 tasks and scalability up to 8 GPUs with 85% parallel efficiency. This work establishes meta-learning as a transformative approach for parametric PDE solving, enabling practical deployment of PINNs in real-time and multi-query scenarios. However, chaotic systems like Kuramoto-Sivashinsky equations present increased challenges, with our best method achieving L2 relative error of 0.089 compared to 0.034 average across all problems, indicating the need for specialized approaches for highly nonlinear dynamics.