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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.00643 |
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| _version_ | 1866909852854386688 |
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| author | Matveev, Albert Ghosh, Sanmitra Hussain, Aamal Leahy, James-Michael Michaelides, Michalis |
| author_facet | Matveev, Albert Ghosh, Sanmitra Hussain, Aamal Leahy, James-Michael Michaelides, Michalis |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00643 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operators Matveev, Albert Ghosh, Sanmitra Hussain, Aamal Leahy, James-Michael Michaelides, Michalis Machine Learning 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. |
| title | Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operators |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2508.00643 |