Trustworthy Few-Shot Learning for Scientific Computing: Meta-Learning Physics-Informed Neural Networks with Reliability Guarantees

Deploying neural networks for scientific computing in high-stakes engineering applications requires trustworthiness guarantees including reliable predictions respecting physical laws, interpretability through physics-based constraints, robustness to distribution shifts across parameter regimes, and computational efficiency for real-time deployment. We present a comprehensive meta-learning framework that enhances trustworthiness of Physics-Informed Neural Networks (PINNs) for parametric partial differential equations through rapid few-shot adaptation while maintaining physical consistency. Our framework introduces four architectures—MetaPINN, PhysicsInformedMetaLearner, TransferLearningPINN, and DistributedMetaPINN—that achieve 79% error reduction (L2: 0.034 vs 0.160) compared to standard PINNs while enabling 6.5× faster adaptation. Critically for trustworthy deployment, physics-informed meta-learning prevents physical constraint violations (0% vs 8.3% for standard deep learning), maintains interpretable physics-based structure, and provides robust few-shot performance (L2: 0.067 in 1-shot vs 0.245 for baselines). Through comprehensive evaluation across seven parametric PDE families including heat transfer, fluid dynamics, and reaction-diffusion systems, we demonstrate that meta-learning with physics constraints simultaneously improves accuracy, reliability, interpretability, and robustness—dimensions that typically trade off in pure data-driven approaches. Break-even analysis establishes cost-effectiveness after 13-16 tasks with 85% parallel efficiency on 8 GPUs, enabling practical deployment in engineering optimization and real-time control requiring trustworthy predictions. Our results provide evidence that combining meta-learning with physics-informed constraints offers a pathway to trustworthy neural networks for scientific computing where failures have significant consequences.