Linearization Turns Neural Operators into Function-Valued Gaussian Processes

Emilia Magnani, Marvin Pförtner, Tobias Weber, Philipp Hennig
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:42421-42452, 2025.

Abstract

Neural operators generalize neural networks to learn mappings between function spaces from data. They are commonly used to learn solution operators of parametric partial differential equations (PDEs) or propagators of time-dependent PDEs. However, to make them useful in high-stakes simulation scenarios, their inherent predictive error must be quantified reliably. We introduce LUNO, a novel framework for approximate Bayesian uncertainty quantification in trained neural operators. Our approach leverages model linearization to push (Gaussian) weight-space uncertainty forward to the neural operator’s predictions. We show that this can be interpreted as a probabilistic version of the concept of currying from functional programming, yielding a function-valued (Gaussian) random process belief. Our framework provides a practical yet theoretically sound way to apply existing Bayesian deep learning methods such as the linearized Laplace approximation to neural operators. Just as the underlying neural operator, our approach is resolution-agnostic by design. The method adds minimal prediction overhead, can be applied post-hoc without retraining the network, and scales to large models and datasets. We evaluate these aspects in a case study on Fourier neural operators.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-magnani25a, title = {Linearization Turns Neural Operators into Function-Valued {G}aussian Processes}, author = {Magnani, Emilia and Pf\"{o}rtner, Marvin and Weber, Tobias and Hennig, Philipp}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {42421--42452}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/magnani25a/magnani25a.pdf}, url = {https://proceedings.mlr.press/v267/magnani25a.html}, abstract = {Neural operators generalize neural networks to learn mappings between function spaces from data. They are commonly used to learn solution operators of parametric partial differential equations (PDEs) or propagators of time-dependent PDEs. However, to make them useful in high-stakes simulation scenarios, their inherent predictive error must be quantified reliably. We introduce LUNO, a novel framework for approximate Bayesian uncertainty quantification in trained neural operators. Our approach leverages model linearization to push (Gaussian) weight-space uncertainty forward to the neural operator’s predictions. We show that this can be interpreted as a probabilistic version of the concept of currying from functional programming, yielding a function-valued (Gaussian) random process belief. Our framework provides a practical yet theoretically sound way to apply existing Bayesian deep learning methods such as the linearized Laplace approximation to neural operators. Just as the underlying neural operator, our approach is resolution-agnostic by design. The method adds minimal prediction overhead, can be applied post-hoc without retraining the network, and scales to large models and datasets. We evaluate these aspects in a case study on Fourier neural operators.} }
Endnote
%0 Conference Paper %T Linearization Turns Neural Operators into Function-Valued Gaussian Processes %A Emilia Magnani %A Marvin Pförtner %A Tobias Weber %A Philipp Hennig %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-magnani25a %I PMLR %P 42421--42452 %U https://proceedings.mlr.press/v267/magnani25a.html %V 267 %X Neural operators generalize neural networks to learn mappings between function spaces from data. They are commonly used to learn solution operators of parametric partial differential equations (PDEs) or propagators of time-dependent PDEs. However, to make them useful in high-stakes simulation scenarios, their inherent predictive error must be quantified reliably. We introduce LUNO, a novel framework for approximate Bayesian uncertainty quantification in trained neural operators. Our approach leverages model linearization to push (Gaussian) weight-space uncertainty forward to the neural operator’s predictions. We show that this can be interpreted as a probabilistic version of the concept of currying from functional programming, yielding a function-valued (Gaussian) random process belief. Our framework provides a practical yet theoretically sound way to apply existing Bayesian deep learning methods such as the linearized Laplace approximation to neural operators. Just as the underlying neural operator, our approach is resolution-agnostic by design. The method adds minimal prediction overhead, can be applied post-hoc without retraining the network, and scales to large models and datasets. We evaluate these aspects in a case study on Fourier neural operators.
APA
Magnani, E., Pförtner, M., Weber, T. & Hennig, P.. (2025). Linearization Turns Neural Operators into Function-Valued Gaussian Processes. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:42421-42452 Available from https://proceedings.mlr.press/v267/magnani25a.html.

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