{"ID":2887988,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.00643","arxiv_id":"2508.00643","title":"Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operators","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.","short_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 sc...","url_abs":"https://arxiv.org/abs/2508.00643","url_pdf":"https://arxiv.org/pdf/2508.00643v2","authors":"[\"Albert Matveev\",\"Sanmitra Ghosh\",\"Aamal Hussain\",\"James-Michael Leahy\",\"Michalis Michaelides\"]","published":"2025-08-01T13:57:19Z","proceeding":"cs.LG","tasks":"[\"cs.LG\"]","methods":"[\"Diffusion Model\"]","has_code":false}