Randomized neural operator for parametric PDEs with fast training and conformal uncertainty quantification

2026-06-28

AI Summary

This work proposes PCA–RaNN, an efficient framework for solving parametric partial differential equations in multi-query settings such as uncertainty quantification, design optimization, and inverse problems. By leveraging principal component analysis for dimensionality reduction, the method reformulates neural operator learning as a linear regression problem based on fixed random features, enabling extremely fast training via closed-form least squares readout and online adaptation through recursive least squares. The approach further introduces an energy-matching scaling rule, lightweight BFGS fine-tuning, and ensemble averaging to construct split conformal prediction intervals, substantially enhancing both training efficiency and uncertainty quantification. Benchmarked on canonical problems—including Burgers’, Darcy, Navier–Stokes, and the inverse heat equation—the method achieves speedups of one to three orders of magnitude while maintaining competitive accuracy.

Abstract

Repeatedly solving parametric PDEs is essential for uncertainty quantification, design optimization and inverse problems, but conventional neural operators require expensive non-convex training. We introduce PCA--RaNN, a randomized latent neural operator that combines PCA-based dimensionality reduction with fixed random features and a closed-form least-squares readout. It recasts latent operator learning as fixed-feature linear regression, reducing training time by one to three orders of magnitude across benchmarks while maintaining competitive accuracy. We introduce an energy-matched scaling rule and a lightweight two-parameter BFGS refinement to correct suboptimal feature scales. Ensemble averaging reduces predictive variance. On Burgers, Darcy, Navier--Stokes and backward heat equation benchmarks, PCA--RaNN provides a favorable speed--accuracy trade-off against operator-learning baselines. The ensemble supports split-conformal prediction intervals, and the linear readout enables rapid online adaptation via recursive least squares without retraining hidden features. This provides an efficient, uncertainty-aware surrogate for many-query scientific workflows.

Problem

• parametric PDEs
• uncertainty quantification
• neural operators
• fast training
• conformal prediction

Innovation

• randomized neural operator
• parametric PDEs
• fixed random features
• conformal uncertainty quantification
• fast training