Diffusion-Based Stochastic Operator Networks for Uncertainty Quantification in Stochastic Partial Differential Equations

2026-05-19 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Engineering & Applied Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, extended

Summary

The Stochastic Operator Network (SON) is a new machine learning framework designed for uncertainty quantification (UQ) in stochastic partial differential equations (SPDEs). It integrates the Deep Operator Network (DeepONet) architecture with Stochastic Neural Networks (SNNs) to enable probabilistic prediction of solution operators directly from noisy data, without requiring explicit knowledge of noise sources. A key innovation is a two-phase training strategy that first trains a deterministic DeepONet to capture the solution's overall structure (Phase I), then uses an SNN to learn and quantify stochastic uncertainty around that pre-trained mean (Phase II). This modular approach enhances training efficiency and stability, making SON applicable to complex SPDE models. Numerical experiments on benchmark SPDEs, including stochastic advection-diffusion, Navier-Stokes, and reaction-diffusion equations, demonstrate SON's accuracy in mean prediction and robustness in quantifying predictive uncertainty, even with noisy training data.

Key takeaway

For AI Scientists and Machine Learning Engineers working with SPDEs and noisy data, SON offers a robust method for uncertainty quantification. Its two-phase training strategy, separating deterministic operator learning from stochastic uncertainty modeling, significantly improves efficiency and stability compared to joint training. You should consider implementing SON to achieve accurate mean predictions and reliable uncertainty estimates in your SPDE solution operators, especially for complex systems where traditional UQ methods are computationally prohibitive.

Key insights

SON combines DeepONet with SNNs and a two-phase training for efficient SPDE uncertainty quantification.

Principles

• Decouple deterministic and stochastic learning for efficiency.
• Stochastic diffusion terms characterize model uncertainty.

Method

SON uses a two-phase training: Phase I trains a DeepONet for mean solution, then Phase II trains an SNN on its output to model stochastic diffusion and quantify uncertainty, optimizing with a Hamiltonian-type loss and Stochastic Maximum Principle.

In practice

• Apply SON for UQ in complex SPDEs with noisy data.
• Use the two-phase training to improve stability and efficiency.

Topics

• Stochastic Partial Differential Equations
• Uncertainty Quantification
• Deep Operator Networks
• Stochastic Neural Networks
• Two-Phase Training Strategy

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

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