Structure-Aware Epistemic Uncertainty Quantification for Neural Operator PDE Surrogates

2026-03-13 · Source: cs.LG updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences, Engineering & Applied Sciences · Depth: Expert, extended

Summary

A new structure-aware epistemic uncertainty quantification (UQ) scheme has been developed for neural operators (NOs) used as PDE surrogates. This method addresses the challenge of NO predictions exhibiting significant epistemic uncertainty due to factors like finite data and distribution shift. Instead of applying unstructured weight perturbations across the entire network, the proposed scheme restricts Monte Carlo sampling to the lifting module, treating the propagation and recovery modules as deterministic. This approach, instantiated with channel-wise multiplicative feature dropout and Gaussian feature perturbation, yields more reliable coverage, tighter uncertainty bands, and improved alignment between residuals and uncertainty. Experiments on challenging PDE benchmarks, including discontinuous-coefficient Darcy flow and geometry-shifted 3D car CFD surrogates, demonstrate its superior performance and practicality compared to baselines like MC Dropout and Deep Ensembles, while maintaining computational efficiency on a single Nvidia GeForce RTX 4090 GPU.

Key takeaway

For AI Scientists and Research Scientists developing or deploying neural operator PDE surrogates, you should consider implementing structure-aware UQ by focusing stochasticity on the lifting module. This approach provides more accurate and spatially faithful uncertainty estimates with tighter bands, enhancing the reliability of your models in scientific computing workflows and reducing the risk of over-conservative interventions.

Key insights

Restricting uncertainty sampling to the lifting module improves neural operator uncertainty quantification efficiency and accuracy.

Principles

• Epistemic uncertainty in NOs is primarily due to limited data and imperfect training.
• Unstructured perturbations can degrade predictive accuracy and inflate uncertainty bands.
• Lifting module perturbations can be seen as initial feature-space uncertainty.

Method

The method injects stochasticity only into the lifting module of neural operators using channel-wise multiplicative feature dropout or Gaussian feature perturbation, followed by deterministic propagation and recovery, to estimate epistemic uncertainty.

In practice

• Use channel-wise multiplicative feature noise for lifted features.
• Apply Gaussian feature perturbation with matched variance.
• Integrate into existing NO architectures with minimal modification.

Topics

• Neural Operators
• Uncertainty Quantification
• Epistemic Uncertainty
• PDE Surrogates
• Feature Perturbation

Best for: AI Scientist, Research Scientist, AI Researcher, Machine Learning Engineer, MLOps Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by cs.LG updates on arXiv.org.