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

Aug 12, 2025	I will present our work on Constructive Disintegration and Conditional Modes at ProbNum 2025.
Jun 6, 2025	I will present our work on uncertainty quantification for neural operators (spotlight) and Laplace approximations in JAX (CODEML workshop) at ICML 2025 in Vancouver, Canada.
Mar 12, 2025	I will present our work on Computation-Aware Kalman Filtering and Smoothing at AISTATS 2025 in Mai Khao, Thailand.

selected publications

Constructive Disintegration and Conditional Modes

Nathaël Da Costa, Marvin Pförtner, and Jon Cockayne

2025

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Conditioning, the central operation in Bayesian statistics, is formalised by the notion of disintegration of measures. However, due to the implicit nature of their definition, constructing disintegrations is often difficult. A folklore result in machine learning conflates the construction of a disintegration with the restriction of probability density functions onto the subset of events that are consistent with a given observation. We provide a comprehensive set of mathematical tools which can be used to construct disintegrations and apply these to find densities of disintegrations on differentiable manifolds. Using our results, we provide a disturbingly simple example in which the restricted density and the disintegration density drastically disagree. Motivated by applications in approximate Bayesian inference and Bayesian inverse problems, we further study the modes of disintegrations. We show that the recently introduced notion of a “conditional mode” does not coincide in general with the modes of the conditional measure obtained through disintegration, but rather the modes of the restricted measure. We also discuss the implications of the discrepancy between the two measures in practice, advocating for the utility of both approaches depending on the modelling context.
@misc{DaCosta2025ConstructiveDisintegration, author = {Da Costa, Natha\"el and Pf\"ortner, Marvin and Cockayne, Jon}, title = {Constructive Disintegration and Conditional Modes}, year = {2025}, archiveprefix = {arXiv}, eprint = {2508.00617}, primaryclass = {cs.LG}, doi = {10.48550/arxiv.2508.00617}, url = {https://arxiv.org/abs/2508.00617}, }
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 {Kalman} Filtering and Smoothing}, editor = {Li, Yingzhen and Mandt, Stephan and Agrawal, Shipra and Khan, Emtiyaz}, booktitle = {Proceedings of the 28th International Conference on Artificial Intelligence and Statistics (AISTATS)}, volume = {258}, series = {Proceedings of Machine Learning Research}, pages = {2071--2079}, publisher = {PMLR}, address = {Mai Khao, Thailand}, year = {2025}, archiveprefix = {arXiv}, eprint = {2405.08971}, primaryclass = {cs.LG}, doi = {10.48550/arxiv.2405.08971}, url = {https://proceedings.mlr.press/v258/pfortner25a.html}, }
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\"ortner, Marvin and Fortuin, Vincent and Hennig, Philipp and Bamler, Robert}, title = {{FSP-Laplace}: Function-Space Priors for the {Laplace} Approximation in {Bayesian} Deep Learning}, editor = {Globerson, A. and Mackey, L. and Belgrave, D. and Fan, A. and Paquet, U. and Tomczak, J. and Zhang, C.}, booktitle = {Advances in Neural Information Processing Systems}, volume = {37}, pages = {13897--13926}, publisher = {Curran Associates, Inc.}, year = {2024}, archiveprefix = {arXiv}, eprint = {2407.13711}, primaryclass = {cs.LG}, 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 {Gaussian} Process Regression Generalizes Linear {PDE} Solvers}, year = {2022}, archiveprefix = {arXiv}, eprint = {2212.12474}, primaryclass = {cs.LG}, doi = {10.48550/arxiv.2212.12474}, url = {https://arxiv.org/abs/2212.12474}, }