Data generation with optimal experimental design for operator learning

Partial differential equations (PDEs) are fundamental to modeling complex physical phenomena across scientific disciplines. While operator learning offers a promising alternative to conventional PDE solvers, it generally requires substantial high-fidelity training data, resulting in significant computational costs. Existing approaches typically sample PDE parameters uniformly or heuristically, which can be inefficient in computational resources and may lead to suboptimal neural operator performance. Leveraging functional encoding, we propose a systematic framework that adapts (finite-dimensional) optimal experimental design (OED) principles for generating informative training datasets while minimizing computational costs. The OED framework employs physics-infused informativeness metrics—solution variance, energy dissipation, and high-frequency spectral content—to guide strategic parameter sampling using Bayesian inference with adaptive acquisition strategies. The framework achieves substantial improvements in data efficiency through computationally lightweight OED operations that incur negligible overhead compared to expensive PDE simulations. We empirically demonstrate that strategic parameter sampling guided by informativeness metrics significantly outperforms uniform random sampling strategies across multiple PDE benchmarks, including Burgers, Darcy, and Navier-Stokes equations. Comprehensive evaluations using both Fourier Neural Operators (FNO) and Deep Operator Networks (DeepONet), two leading neural operator architectures, confirm the generality of the approach. Using equivalent computational budgets, the method achieves substantially lower validation errors by concentrating simulations on parameter regions that provide maximum learning value. We provide a principled yet practical approach to training data generation that reduces the computational barrier to deploying neural operators for complex parametric PDEs. We release an open-source implementation to make this data-efficient operator learning framework accessible.