Geometry, Algebra and Physics in Deep Neural Networks

Daniel Persson gave a talk on Math for AI vs AI for Math at the online seminar of MaScAmp, the ERC Synergy project on the Mathematics of Scattering Amplitudes, on November 2, 2026. The talk covers both directions: how symmetry shapes the design of neural networks, and how AI agents can assist mathematical research. Slides are available here.

Daniel Persson and collaborators have been granted 2 million SEK over two years by Vinnova, Sweden’s innovation agency, for the project Ultrasound training for acute trauma care using AI. eFAST ultrasound lets trauma teams spot life-threatening conditions such as internal bleeding within minutes, but acquiring good images requires a lot of supervised practice. The project will build and evaluate an AI-supported eFAST training coach for trauma care. It is a collaboration between Sahlgrenska University Hospital, Chalmers Next Labs, Sindrion Technologies and the Department of Mathematical Sciences at Chalmers, and builds on the training platform developed at Kymeon Medical Technologies, a spin-out from Sindrion. Collaborators include Carl Hallgren (Sahlgrenska), Jonna Stålring Westerberg (Chalmers Next Labs) and Sebastian Bramberger (Kymeon).

Daniel Persson has been awarded a project grant of 4.2 million SEK over four years from the Swedish Research Council (Vetenskapsrådet) for the project Invariant Theory and Geometric Deep Learning. The aim is to use invariant theory to build new non-linear equivariant architectures for deep learning. The project builds on the paper Equivariant non-linear maps for neural networks on homogeneous spaces by Elias Nyholm, Oscar Carlsson, Maurice Weiler and Daniel Persson, and on the wider research of the group.

Jan Gerken co-organized the Wallenberg Advanced Scientific Forum – Autumn 2026: Symmetries in Neural Networks together with Axel Flinth, Elisenda Grigsby, Fredrik Kahl, Kathlén Kohn and Fredrik Ohlsson. The highly interactive workshop took place September 29 – October 2, 2026 at Rånäs Castle and brought together experts to study symmetries in data, parameters and training dynamics of neural networks. Elias Nyholm and Longde Huang also participated in the forum.

Public lecture by Daniel Persson for high school students at Vadsbogymnasiet in Mariestad. Slides (in Swedish) are available here.

The new CHAIR theme Mathematical Foundations of AI has launched, with Daniel Persson and Jan Gerken among its co-organizers. The theme brings together researchers from computer science, mathematics, physics and electrical engineering at Chalmers and the University of Gothenburg to develop the theoretical foundations of AI, through regular seminars, community retreats and international workshops. More information is available on the CHAIR website.

Miaowen Dong, Longde Huang, Hampus Linander, and Philipp Misof presented their recent work at the third GeUmetric Deep Learning Workshop in Umeå. This conference, co-organized by Jan Gerken, brought together researchers with interests in the mathematics, geometry and physics behind deep learning.

A new preprint Boosting Data Augmentation with Stochastic Weight Averaging by Longde Huang, Axel Flinth and Jan Gerken is now available on the arXiv. The paper studies stochastic weight averaging (SWA) as a cheap alternative to deep ensembles for making augmented models equivariant. By approximating the training trajectory at the end of training with an Ornstein–Uhlenbeck process, the paper shows that in the infinite-width limit SWA on augmented data yields an equivariance boost beyond what the performance gain from SWA alone would suggest, confirmed by extensive experiments on computer vision and graph classification models with discrete and continuous symmetries.

Daniel Persson gave a talk on Beyond the chatbot - How I use AI in my mathematical research at the workshop New connections between physics and number theory, held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, July 13–17, 2026. Slides are available here.

Jan Gerken and Daniel Persson have been awarded a Proof-of-Concept grant from the Knut and Alice Wallenberg Foundation for the project AI-Assisted Ultrasound Guidance for Scalable Medical Training. This is a joint project with Carl Hallgren at Sahlgrenska University Hospital and Sebastian Bramberger at Sindrion Technologies. The Proof-of-Concept program bridges the gap between basic academic research and innovation, giving research groups the opportunity to develop early discoveries towards validated methods, products or processes and to prepare them for commercialization.

Jan Gerken presented our paper Finite-Width Neural Tangent Kernels from Feynman Diagrams, joint work with Max Guillen and Philipp Misof, as a poster at ICML 2026 in Seoul. The poster is available here.

A new preprint Equivariance and Augmentation for Bayesian Neural Networks by Miaowen Dong, Axel Flinth and Jan Gerken is now available on the arXiv. The paper studies data augmentation for Bayesian neural networks trained with variational inference, deriving conditions under which exact equivariance is reached, bounds on the equivariance error, and three novel symmetrization techniques. One of them, orbit expansion, outperforms the baseline in both equivariance and overall performance.

This is the first paper for our PhD student Miaowen Dong — congratulations, Miaowen!

Congratulations to our master’s students, who have all successfully defended their theses:

• Tage Tykesson: Deep Learning Climate Model Emulation with PEAR
• Atharva Khandait: Discrete Geometry for Comparing and Transferring Neural Representations
• Pietro Rosso: Equivariant Inductive Biases for Weather Prediction with PEAR
• Jonatan Larsen and Filip Nyman: SÄLEN and the Art of Metal Binding

Well done to all of them!

Daniel Persson gave an invited talk on Geometric Deep Learning - From equivariance to weather predictions at the Mathematical Foundations of AI workshop, held at Chalmers University of Technology (Palmstedtsalen), Gothenburg, June 8–10, 2026. Slides are available here.

A substantially updated version of our paper PEAR: Equal Area Weather Forecasting on the Sphere by Hampus Linander, Christoffer Petersson, Daniel Persson and Jan Gerken is now available on the arXiv. The new version shows that PEAR outperforms a much wider range of baseline architectures than previously evaluated, demonstrates that PEAR performs well in climate model emulation, and presents new results on the symmetry properties of PEAR.

The new results on symmetry properties and climate modeling are thanks to the work of our master’s students Pietro Rosso and Tage Tykesson. Congratulations to both!

Daniel Persson is interviewed in the Swedish popular science magazine Forskning & Framsteg about the recent AI breakthrough in mathematics, where an AI model solved the 80-year-old Erdős unit distance problem. Daniel comments: “It is a very exciting breakthrough! It is remarkable that AI has been able to solve a problem at this level, with such advanced mathematics.”

A new preprint From Layers to Networks: Comparing Neural Representations via Diffusion Geometry is now available on the arXiv. The paper brings tools from diffusion geometry and multi-view learning to the comparison of neural representations, showing that a broad class of representational similarity measures can be reformulated via row-stochastic Markov matrices. This yields multi-scale variants of Centered Kernel Alignment and Distance Correlation and enables network-to-network comparisons, achieving state-of-the-art results on the Representational Similarity (ReSi) benchmark.

First author is our master’s student Atharva Khandait, together with Jan Gerken. Congratulations to Atharva on his first publication!

When Philipp Misof visited last year’s LOGML Summer School in London, the week-long group project sparked an idea for a new paper using Diffusion Geometry to learn geometric differential-forms from point clouds. This work was done together with Bruno Trentini, Jacob Hume, Vincenzo Antonio Isoldi, Katya Ivshina and Kelly Maggs and is now available as a preprint.

Daniel Persson, together with collaborators Emma Andersdotter and Fredrik Ohlsson at Umeå University, have published a new preprint on Steerable Neural ODEs on Homogeneous Spaces. The paper introduces a novel geometric framework for equivariant neural ODEs on homogeneous spaces, using parallel transport to steer feature vectors transforming under local symmetry groups.

Max Guillen and Jan Gerken have published a new preprint on Criticality and Saturation in Orthogonal Neural Networks. The paper derives layer-wise recursion relations for the finite-width statistics of networks with orthogonal weight initialization and extends the recently-introduced Feynman diagram framework to this setting, providing a theoretical explanation for the depth-stability of orthogonally-initialized nonlinear networks.