Marvin Pförtner

I am a PhD student in Philipp Hennig’s group at the University of Tübingen and the International Max Planck Research School for Intelligent Systems (IMPRS-IS). My research interests lie at the intersection of Bayesian machine learning and numerical analysis. More specifically, my work revolves around

algorithms for scalable (approximate) Gaussian process inference,
Gaussian process theory (sample path properties, Gaussian measure theory),
probabilistic numerical methods for partial differential equations, and
Bayesian deep learning with Laplace approximations.

I’m also interested in applications of all the above to scientific inference tasks.

I like to tackle problems using the framework of matrix-free (probabilistic) numerical linear algebra, which often leads to elegant and efficient algorithms.

news

Mar 12, 2025	I will present our work on Computation-Aware Kalman Filtering and Smoothing at AISTATS 2025 in Mai Khao, Thailand.
Jan 2, 2025	I will give a talk on “Probabilistic Functional Programming” at the MFO Workshop 2505 on “Overparametrization, Regularization, Identifiability and Uncertainty in Machine Learning” in Oberwolfach, Germany.
Dec 1, 2024	I will be attending NeurIPS 2024 in Vancouver.

selected publications

Computation-Aware Kalman Filtering and Smoothing

Marvin Pförtner, Jonathan Wenger, Jon Cockayne, and Philipp Hennig

In Proceedings of the 28th International Conference on Artificial Intelligence and Statistics (AISTATS), 2025

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Kalman filtering and smoothing are the foundational mechanisms for efficient inference in Gauss-Markov models. However, their time and memory complexities scale prohibitively with the size of the state space. This is particularly problematic in spatiotemporal regression problems, where the state dimension scales with the number of spatial observations. Existing approximate frameworks leverage low-rank approximations of the covariance matrix. Since they do not model the error introduced by the computational approximation, their predictive uncertainty estimates can be overly optimistic. In this work, we propose a probabilistic numerical method for inference in high-dimensional Gauss-Markov models which mitigates these scaling issues. Our matrix-free iterative algorithm leverages GPU acceleration and crucially enables a tunable trade-off between computational cost and predictive uncertainty. Finally, we demonstrate the scalability of our method on a large-scale climate dataset.
@inproceedings{Pfoertner2025CAKF, author = {Pf\"ortner, Marvin and Wenger, Jonathan and Cockayne, Jon and Hennig, Philipp}, title = {Computation-Aware {K}alman Filtering and Smoothing}, year = {2025}, booktitle = {Proceedings of the 28th International Conference on Artificial Intelligence and Statistics (AISTATS)}, publisher = {PMLR}, volume = {258}, doi = {10.48550/arxiv.2405.08971}, url = {https://arxiv.org/abs/2405.08971}, }
FSP-Laplace: Function-Space Priors for the Laplace Approximation in Bayesian Deep Learning

Tristan Cinquin, Marvin Pförtner, Vincent Fortuin, Philipp Hennig, and Robert Bamler

In Advances in Neural Information Processing Systems, 2024

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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.
@inproceedings{Cinquin2024FSPLaplace, author = {Cinquin, Tristan and Pf{\"o}rtner, Marvin and Fortuin, Vincent and Hennig, Philipp and Bamler, Robert}, title = {FSP-Laplace: Function-Space Priors for the Laplace Approximation in Bayesian Deep Learning}, year = {2024}, booktitle = {Advances in Neural Information Processing Systems}, volume = {37}, editor = {Globerson, A. and Mackey, L. and Belgrave, D. and Fan, A. and Paquet, U. and Tomczak, J. and Zhang, C.}, publisher = {Curran Associates, Inc.}, pages = {13897--13926}, doi = {10.48550/arxiv.2407.13711}, url = {https://papers.neurips.cc/paper_files/paper/2024/hash/19774ce2d4b0d17a3a8aea26ad99fe8a-Abstract-Conference.html}, }
Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers

Marvin Pförtner, Ingo Steinwart, Philipp Hennig, and Jonathan Wenger

2022

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Linear partial differential equations (PDEs) are an important, widely applied class of mechanistic models, describing physical processes such as heat transfer, electromagnetism, and wave propagation. In practice, specialized numerical methods based on discretization are used to solve PDEs. They generally use an estimate of the unknown model parameters and, if available, physical measurements for initialization. Such solvers are often embedded into larger scientific models with a downstream application and thus error quantification plays a key role. However, by ignoring parameter and measurement uncertainty, classical PDE solvers may fail to produce consistent estimates of their inherent approximation error. In this work, we approach this problem in a principled fashion by interpreting solving linear PDEs as physics-informed Gaussian process (GP) regression. Our framework is based on a key generalization of the Gaussian process inference theorem to observations made via an arbitrary bounded linear operator. Crucially, this probabilistic viewpoint allows to (1) quantify the inherent discretization error; (2) propagate uncertainty about the model parameters to the solution; and (3) condition on noisy measurements. Demonstrating the strength of this formulation, we prove that it strictly generalizes methods of weighted residuals, a central class of PDE solvers including collocation, finite volume, pseudospectral, and (generalized) Galerkin methods such as finite element and spectral methods. This class can thus be directly equipped with a structured error estimate. In summary, our results enable the seamless integration of mechanistic models as modular building blocks into probabilistic models by blurring the boundaries between numerical analysis and Bayesian inference.
@misc{Pfoertner2022LinPDEGP, author = {Pf\"ortner, Marvin and Steinwart, Ingo and Hennig, Philipp and Wenger, Jonathan}, title = {Physics-Informed {G}aussian Process Regression Generalizes Linear {PDE} Solvers}, year = {2022}, publisher = {arXiv}, doi = {10.48550/arxiv.2212.12474}, url = {https://arxiv.org/abs/2212.12474}, }
Posterior and Computational Uncertainty in Gaussian Processes

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

In Advances in Neural Information Processing Systems, 2022

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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.
@inproceedings{Wenger2022IterGP, author = {Wenger, Jonathan and Pleiss, Geoff and Pf\"ortner, Marvin and Hennig, Philipp and Cunningham, John P.}, title = {Posterior and Computational Uncertainty in {G}aussian Processes}, year = {2022}, booktitle = {Advances in Neural Information Processing Systems}, volume = {35}, editor = {Koyejo, S. and Mohamed, S. and Agarwal, A. and Belgrave, D. and Cho, K. and Oh, A.}, publisher = {Curran Associates, Inc.}, pages = {10876--10890}, doi = {10.48550/arxiv.2205.15449}, url = {https://proceedings.neurips.cc/paper_files/paper/2022/hash/4683beb6bab325650db13afd05d1a14a-Abstract-Conference.html}, }