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Marvin Pförtner

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4 papers

2 author rows

ICML Conference 2025 Conference Paper

Linearization Turns Neural Operators into Function-Valued Gaussian Processes

Emilia Magnani
Marvin Pförtner
Tobias Weber
Philipp Hennig

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.

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NeurIPS Conference 2024 Conference Paper

FSP-Laplace: Function-Space Priors for the Laplace Approximation in Bayesian Deep Learning

Tristan Cinquin
Marvin Pförtner
Vincent Fortuin
Philipp Hennig
Robert Bamler

Laplace approximations are popular techniques for endowing deep networks with epistemic uncertainty estimates as they can be applied without altering the predictions of the trained network, and they scale to large models and datasets. While the choice of prior strongly affects the resulting posterior distribution, computational tractability and lack of interpretability of the weight space typically limit the Laplace approximation to isotropic Gaussian priors, which are known to cause pathological behavior as depth increases. As a remedy, we directly place a prior on function space. More precisely, since Lebesgue densities do not exist on infinite-dimensional function spaces, we recast training as finding the so-called weak mode of the posterior measure under a Gaussian process (GP) prior restricted to the space of functions representable by the neural network. Through the GP prior, one can express structured and interpretable inductive biases, such as regularity or periodicity, directly in function space, while still exploiting the implicit inductive biases that allow deep networks to generalize. After model linearization, the training objective induces a negative log-posterior density to which we apply a Laplace approximation, leveraging highly scalable methods from matrix-free linear algebra. Our method provides improved results where prior knowledge is abundant (as is the case in many scientific inference tasks). At the same time, it stays competitive for black-box supervised learning problems, where neural networks typically excel.

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NeurIPS Conference 2024 Conference Paper

Reparameterization invariance in approximate Bayesian inference

Hrittik Roy
Marco Miani
Carl Henrik Ek
Philipp Hennig
Marvin Pförtner
Lukas Tatzel
Søren Hauberg

Current approximate posteriors in Bayesian neural networks (BNNs) exhibit a crucial limitation: they fail to maintain invariance under reparameterization, i. e. BNNs assign different posterior densities to different parametrizations of identical functions. This creates a fundamental flaw in the application of Bayesian principles as it breaks the correspondence between uncertainty over the parameters with uncertainty over the parametrized function. In this paper, we investigate this issue in the context of the increasingly popular linearized Laplace approximation. Specifically, it has been observed that linearized predictives alleviate the common underfitting problems of the Laplace approximation. We develop a new geometric view of reparametrizations from which we explain the success of linearization. Moreover, we demonstrate that these reparameterization invariance properties can be extended to the original neural network predictive using a Riemannian diffusion process giving a straightforward algorithm for approximate posterior sampling, which empirically improves posterior fit.

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NeurIPS Conference 2022 Conference Paper

Posterior and Computational Uncertainty in Gaussian Processes

Jonathan Wenger
Geoff Pleiss
Marvin Pförtner
Philipp Hennig
John P. Cunningham

Gaussian processes scale prohibitively with the size of the dataset. In response, many approximation methods have been developed, which inevitably introduce approximation error. This additional source of uncertainty, due to limited computation, is entirely ignored when using the approximate posterior. Therefore in practice, GP models are often as much about the approximation method as they are about the data. Here, we develop a new class of methods that provides consistent estimation of the combined uncertainty arising from both the finite number of data observed and the finite amount of computation expended. The most common GP approximations map to an instance in this class, such as methods based on the Cholesky factorization, conjugate gradients, and inducing points. For any method in this class, we prove (i) convergence of its posterior mean in the associated RKHS, (ii) decomposability of its combined posterior covariance into mathematical and computational covariances, and (iii) that the combined variance is a tight worst-case bound for the squared error between the method's posterior mean and the latent function. Finally, we empirically demonstrate the consequences of ignoring computational uncertainty and show how implicitly modeling it improves generalization performance on benchmark datasets.

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