publications

2026

1. Information Geometric Regularization of the Barotropic Euler Equation

   Ruijia Cao and Florian Schäfer

   SIAM Journal on Scientific Computing, 2026

   DOI arXiv Blog
2. Möbius Inversion and the Iterated Bootstrap

   Florian Schäfer

   SIAM Journal on Mathematics of Data Science, 2026

   arXiv
3. Information geometric regularization for computing sensitivities of flows with shocks

   Florian Schäfer

   arXiv preprint arXiv:2608.09759, 2026

   arXiv
4. Shocks without shock capturing: Information geometric regularization of finite volume methods for Navier–Stokes-like problems

   Anand Radhakrishnan, Benjamin Wilfong, Spencer H Bryngelson, and Florian Schäfer

   Computer Methods in Applied Mechanics and Engineering, 2026

   arXiv
5. Discontinuous Galerkin Semidiscretization of the Information Geometric Regularized Compressible Euler Equations

   Brook Eyob, Jesus Arias, Spencer H Bryngelson, and Florian Schäfer

   arXiv preprint arXiv:2608.02223, 2026

   arXiv
6. Shock solutions for the one-dimensional information geometric regularization of compressible flow

   William Barham, Brian K Tran, Ben S Southworth, and Florian Schäfer

   arXiv preprint arXiv:2607.12693, 2026

   arXiv

2025

1. Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes

   Yifan Chen, Houman Owhadi, and Florian Schäfer

   Mathematics of Computation, 2025

   arXiv
2. Optimal Krylov On Average

   Qi Luo and Florian Schäfer

   2025

   arXiv
3. Maximum likelihood discretization of the transport equation

   Brook Eyob and Florian Schäfer

   2025

   arXiv
4. Lightning-fast Boundary Element Method

   Jiong Chen, Florian T Schäfer, and Mathieu Desbrun

   ACM Transactions on Graphics, 2025

   arXiv
5. Hamiltonian Information Geometric Regularization of the Compressible Euler Equations

   William Barham, Brian K Tran, Ben S Southworth, and Florian Schäfer

   2025

   arXiv
6. A Bootstrap Perspective on Stochastic Gradient Descent

   Hongjian Lan, Yucong Liu, and Florian Schäfer

   2025

   arXiv
7. Simulating many-engine spacecraft: Exceeding 1 quadrillion degrees of freedom via information geometric regularization

   Benjamin Wilfong, Anand Radhakrishnan, Henry Le Berre, Daniel Vickers, Tanush Prathi, Nikolaos Tselepidis, Benedikt Dorschner, Reuben Budiardja, Brian Cornille, Stephen Abbott, Florian Schäfer, and Spencer Bryngelson

   In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, , 2025

   Abs DOI arXiv

   We present an optimized implementation of the recently proposed information geometric regularization (IGR) for unprecedented scale simulation of compressible fluid flows applied to multi-engine spacecraft boosters. We improve upon state-of-the-art computational fluid dynamics (CFD) techniques in terms of computational cost, memory footprint, and energy-to-solution metrics. Unified memory on coupled CPU–GPU or APU platforms increases problem size with negligible overhead. Mixed half/single-precision storage and computation are used on well-conditioned numerics. We simulate flow at 200 trillion grid points and 1 quadrillion degrees of freedom, exceeding the current record by a factor of 20. A factor of 4 wall-time speedup is achieved over optimized baselines. Ideal weak scaling is observed on OLCF Frontier, LLNL El Capitan, and CSCS Alps using the full systems. Strong scaling is near ideal at extreme conditions, including 80% efficiency on CSCS Alps with an 8 node baseline and stretching to the full system.

2024

1. Sparse recovery of elliptic solvers from matrix-vector products

   Florian Schäfer and Houman Owhadi

   SIAM Journal on Scientific Computing, 2024

   arXiv
2. Fast Macroscopic Forcing Method

   Spencer H Bryngelson, Florian Schäfer, Jessie Liu, and Ali Mani

   Journal of Computational Physics, 2024

   arXiv
3. Lightning-fast Method of Fundamental Solutions

   Jiong Chen, Florian Schäfer, and Mathieu Desbrun

   ACM Transactions on Graphics (TOG), 2024

   PDF
4. Information geometric regularization of unidimensional pressureless Euler equations yields global strong solutions

   Ruijia Cao and Florian Schäfer

   2024

   arXiv
5. Score Operator Newton transport

   Nisha Chandramoorthy, Florian T Schaefer, and Youssef M Marzouk

   In International Conference on Artificial Intelligence and Statistics, 2024

   arXiv

2023

1. Scalable Bayesian transport maps for high-dimensional non-Gaussian spatial fields

   Matthias Katzfuss and Florian Schäfer

   Journal of the American Statistical Association, 2023
2. Variational sparse inverse Cholesky approximation for latent Gaussian processes via double Kullback-Leibler minimization

   Jian Cao, Myeongjong Kang, Felix Jimenez, Huiyan Sang, Florian Tobias Schaefer, and Matthias Katzfuss

   In International Conference on Machine Learning, 2023

   arXiv
3. Sparse Cholesky factorization by greedy conditional selection

   Stephen Huan, Joseph Guinness, Matthias Katzfuss, Houman Owhadi, and Florian Schäfer

   2023

   arXiv
4. Targeted computation of nonlocal closure operators via an adjoint-based macroscopic forcing method

   Jessie Liu, Florian Schäfer, Spencer H Bryngelson, Tamer A Zaki, and Ali Mani

   2023

   arXiv

2022

1. ZerO Initialization: Initializing Neural Networks with only Zeros and Ones

   Jiawei Zhao, Florian Schäfer, and Anima Anandkumar

   Transactions on Machine Learning Research, 2022
2. Competitive Physics Informed Networks

   Qi Zeng, Yash Kothari, Spencer H Bryngelson, and Florian Tobias Schaefer

   In The Eleventh International Conference on Learning Representations, 2022

2021

1. Robust Reinforcement Learning: A Constrained Game-theoretic Approach

   Jing Yu, Clement Gehring, Florian Schäfer, and Animashree Anandkumar

   In Proceedings of the 3rd Conference on Learning for Dynamics and Control, 2021

   Abs PDF

   Deep reinforcement learning (RL) methods provide state-of-art performance in complex control tasks. However, it has been widely recognized that RL methods often fail to generalize due to unaccounted uncertainties. In this work, we propose a game theoretic framework for robust reinforcement learning that comprises many previous works as special cases. We formulate robust RL as a constrained minimax game between the RL agent and an environmental agent which represents uncertainties such as model parameter variations and adversarial disturbances. To solve the competitive optimization problems arising in our framework, we propose to use competitive mirror descent (CMD). This method accounts for the interactive nature of the game at each iteration while using Bregman divergences to adapt to the global structure of the constraint set. We demonstrate an RRL policy gradient algorithm that leverages Lagrangian duality and CMD. We empirically show that our algorithm is stable for large step sizes, resulting in faster convergence on linear quadratic games.

2. Sparse Cholesky factorization by Kullback-Leibler minimization

   Florian Schäfer, Matthias Katzfuss, and Houman Owhadi

   SIAM Journal on Scientific Computing, 2021

   arXiv
3. Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity

   Florian Schäfer, T. J. Sullivan, and Houman Owhadi

   Multiscale Model. Simul., 2021

   DOI arXiv Blog
4. Polymatrix Competitive Gradient Descent

   Jeffrey Ma, Alistair Letcher, Florian Schäfer, Yuanyuan Shi, and Anima Anandkumar

   2021

   arXiv
5. Multiscale cholesky preconditioning for ill-conditioned problems

   Jiong Chen, Florian Schäfer, Jin Huang, and Mathieu Desbrun

   ACM Transactions on Graphics (TOG), 2021

2020

1. Competitive Mirror Descent

   Florian Schäfer, Anima Anandkumar, and Houman Owhadi

   2020

   arXiv
2. Implicit competitive regularization in GANs

   Florian Schäfer, Hongkai Zheng, and Anima Anandkumar

   In the 37th International Conference on Machine Learning (ICML 2020), 2020

   arXiv Blog Slides

2019

1. Statistical numerical approximation

   Houman Owhadi, Clint Scovel, and Florian Schäfer

   Notices of the American Mathematical Society, 2019

   HTML
2. Competitive gradient descent

   Florian Schäfer and Anima Anandkumar

   Advances in Neural Information Processing Systems, 2019

   arXiv Blog

2018

1. Image extrapolation for the time discrete metamorphosis model – existence and applications

   Alexander Effland, Martin Rumpf, and Florian Schäfer

   SIAM Journal on Imaging Sciences, 2018

   arXiv

2017

1. Time discrete extrapolation in a Riemannian space of images

   Alexander Effland, Martin Rumpf, and Florian Schäfer

   In International Conference on Scale Space and Variational Methods in Computer Vision, 2017