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

• URL: http://arxiv.org/abs/2508.00643v1
• Date: Fri, 01 Aug 2025 13:57:19 GMT
• Title: Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operators
• Authors: Albert Matveev, Sanmitra Ghosh, Aamal Hussain, James-Michael Leahy, Michalis Michaelides,
• Abstract summary: We introduce DINOZAUR: a diffusion-based neural operator parametrization with uncertainty quantification.<n>Our method achieves competitive or superior performance across several PDE benchmarks.
• Score: 1.9689888982532262
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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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