Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operators

2025/08/01 by A. A. Matveev, Sanmitra Ghosh, Matveev, Albert +6 · 1 citation
Computer Science · Decision Sciences · Physics and Astronomy · #FOS: Computer and information sciences #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Numerical Methods and Algorithms #Probabilistic and Robust Engineering Design

paper · pdf · doi:10.48550/arxiv.2508.00643

openalex publication_date 2025/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Operator learning is a powerful paradigm for solving partial differential equations, with Fourier Neural Operators serving as a widely adopted foundation. However, FNOs face significant scalability challenges due to overparameterization and offer no native uncertainty quantification -- a key requirement for reliable scientific and engineering applications. Instead, neural operators rely on post hoc UQ methods that ignore geometric inductive biases. In this work, we introduce DINOZAUR: a diffusion-based neural operator parametrization with uncertainty quantification. Inspired by the structure of the heat kernel, DINOZAUR replaces the dense tensor multiplier in FNOs with a dimensionality-independent diffusion multiplier that has a single learnable time parameter per channel, drastically reducing parameter count and memory footprint without compromising predictive performance. By defining priors over those time parameters, we cast DINOZAUR as a Bayesian neural operator to yield spatially correlated outputs and calibrated uncertainty estimates. Our method achieves competitive or superior performance across several PDE benchmarks while providing efficient uncertainty quantification.

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