Markos Katsoulakis

Ph.D. Brown University, 1993

B.A. National and Kapodistrian University, 1987

Generative Modeling, Scientific Machine Learning, Information Theory, Uncertainty Quantification, Multi-scale Methods

Math 456.1 - Mathematical Modeling: Mathematics of Generative Artificial Intelligence

We learn how to build, use, and critique mathematical models. In modeling we translate scientific questions into mathematical language and corresponding computational methods, and thereby we aim to explain the scientific phenomena under investigation. In the Fall of 2026, we will focus on modeling through Machine Learning and in particular on Generative Artificial Intelligence models. During the last decade, generative models have produced breakthrough results in a wide area of applications, including image generation, text and speech synthesis, applications in science and engineering such as surrogate and sub-grid scale simulators (e.g., in aerospace, atmosphere-ocean science, and materials), and the discovery of new molecules and proteins for drug design, to name only a few. In this course, we will cover, to varying degrees, some of the main families of generative models and related tools, including background on Probability, Estimation, and Information Theory, as well as generative methods such as Normalizing Flows, Generative Adversarial Networks, Energy-based Models, Deep Autoregressive Models, Probabilistic Diffusion Models, and Flow Matching methods. We will focus on their mathematical & statistical foundations and discuss some prominent generative models and related tools. We will also discuss the assessment of generative models and challenges related to trustworthiness, memorization, generalization, and prediction.

Students will work on individual problems related to machine learning modeling and will report their findings to the class in a final presentation. Students can access the class, a detailed syllabus, slides for all lectures, and other course material through Canvas (see our class SPIRE).

1. Y-C. Cheng, H. Gu, T. O. McDonald, W. Wu, S. Tripathi, C. Guarducci, D. Russo, D. L. Abravanel, M. Bailey, Y. Wang, Y. Zhang, Y. Pantazis, H. Levine, R. Jeselsohn, M. A. Katsoulakis, F. Michor, PROFET predicts continuous gene expression dynamics from scRNA-seq data to elucidate heterogeneity of cancer treatment responses, Cell Systems, 2026.
2. K. Kan, X. Li, B. J Zhang, T. Sahai, S. J. Osher, M. A.  Katsoulakis, Optimal Control for Transformer Architectures: Enhancing Generalization, Robustness and Efficiency, to appear 39th Annual Conference on Neural Information Processing Systems, NeurIPS 2025
3. B. J. Zhang, S. Liu, S. J. Osher, M. A. Katsoulakis, Probabilistic operator learning: generative modeling and uncertainty quantification for foundation models of differential equations, preprint.
4. Z. Chen, H. Gu, M. A Katsoulakis, L. Rey-Bellet, W. Zhu, Robust Generative Learning with Lipschitz-Regularized α-Divergences Allows Minimal Assumptions on Target Distributions, to appear, Information and Inference: A Journal of the IMA, (2025)
5. N. Mimikos-Stamatopoulos, B. J. Zhang, M. A. Katsoulakis, Score-based generative models are provably robust: an Uncertainty Quantification perspective, 38th Annual Conference on Neural Information Processing Systems, NeurIPS 2024
6. H. Gu, P. Birmpa, Y. Pantazis, M. A. Katsoulakis, and L. Rey-Bellet, Lipschitz-regularized gradient flows and generative particle algorithms for high-dimensional scarce data, SIAM Data Science,  (2024).
7. B. J. Zhang, M. A. Katsoulakis, A mean-field games laboratory for generative modeling, preprint
8. Z. Chen, M. A. Katsoulakis, L. Rey-Bellet, W.  Zhu, Sample Complexity of Probability Divergences under Group Symmetry), International Conference on Machine Learning ICML 2023
9. J. Birrell, P. Dupuis, M. A. Katsoulakis, Y. Pantazis, L. Rey-Bellet, Function-space regularized Rényi divergences, International Conference on Learning Representations ICLR 2023
10. J. Birrell, P. Dupuis, M. A. Katsoulakis, Y. Pantazis, L. Rey-Bellet, (f, Γ) -Divergences: Interpolating between f-Divergences and Integral Probability Metrics, Thirty-sixth Conference on Neural Information Processing Systems NeurIPS 2022
11. J. Birrell,  M. A. Katsoulakis, L. Rey-Bellet, W. Zhu, Structure-preserving GANs, 39th International Conference on Machine Learning ICML 2022
12. P. Birmpa, J. Feng, M. A. Katsoulakis, L. Rey-Bellet, Model Uncertainty and Correctability for Directed Graphical Models, SIAM/ASA Journal on Uncertainty Quantification 10 (4),1461-1512, (2022)
13. Y. Pantazis, D. Paul, M. Fasoulakis, Y. Stylianou, M. A. Katsoulakis, Cumulant GAN, IEEE Transactions on Neural Networks and Learning Systems, (2022)
14. J. Birrell, M. A. Katsoulakis, Y. Pantazis, Optimizing variational representations of divergences and accelerating their statistical estimation, IEEE Transactions on Information Theory, (2022)
15. J. Birrell, P. Dupuis, M. A. Katsoulakis, Y. Pantazis, L. Rey-Bellet, (f, Γ) -Divergences: Interpolating between f-Divergences and Integral Probability Metrics,  Journal of Machine Learning Research, 23 (39), 1-70, (2022).
16. Eric J. Hall, Søren Taverniers, Markos A. Katsoulakis, Daniel M. Tartakovsky, GINNs: Graph Informed Neural Networks for multiscale physics, J. Comp. Phys. Volume 433, 110192 (2021).
17. Feng J, Lansford J, Katsoulakis M, Vlachos D. Explainable and trustworthy artificial intelligence for correctable modeling in chemical sciences, Science Advances. 2020 October 14; 6(42):eabc3204
18. P. Dupuis, M. A. Katsoulakis, Y. Pantazis, L. Rey-Bellet Sensitivity Analysis for Rare Events based on Renyi Divergence, Annals Applied Prob., 2020 August; 30(4):1507-1533.
19. Gourgoulias K, Katsoulakis M, Rey-Bellet L, Wang J. How Biased Is Your Model? Concentration Inequalities, Information and Model Bias, IEEE Transactions on Information Theory. 2020; 66(5):3079-3097.
20. P. Vilanova, M. A. Katsoulakis, Data-driven, variational model reduction of high-dimensional reaction networks, J. Comp. Phys. Volume 401, 108997 (2020)
21. V. Harmandaris, E. Kalligiannaki, M. A. Katsoulakis, P. Plechac. Path-space Variational Inference for non-equilibrium coarse-grained systems, J. Comp. Phys. 314,355-383 (2016).
22. P. Dupuis, M. A. Katsoulakis, Y. Pantazis, P. Plechac. Path-Space Information Bounds for Uncertainty Quantification and Sensitivity Analysis of Stochastic Dynamics, SIAM/ASA J. Uncertainty Quantification, 4(1), 80-111, (2016).
23. J. E. Sutton, W. Guo, M. A. Katsoulakis & D. G. Vlachos. Effects of correlated parameters and uncertainty in electronic-structure-based chemical kinetic modelling, Nature Chemistry 8, 331–337 (2016).
24. G. Arampatzis, M. A. Katsoulakis and P. Plechac,  Parallelization, processor communication and error analysis in lattice kinetic Monte Carlo, SIAM Num. Analysis, 52, no. 3, 11561182, (2014).
25. Y. Pantazis, D.G. Vlachos and M. A. Katsoulakis, Parametric Sensitivity Analysis for Biochemical Reaction Networks based on Pathwise Information Theory, BMC Bionformatics 14:311, (2013).
26. G. Arampatzis, M. A. Katsoulakis, P. Plechác, M. Taufer, L. Xu. Hierarchical fractional step approximations and parallel kinetic Monte Carlo algorithms. J. Comp. Phys. 231, 7795-7814, (2012).
27. J. Feng ,  M. A. Katsoulakis, A Hamilton Jacobi theory for controlled gradient flows in infinite dimensions, Arch. Rat. Mech. Analysis 192, 2 , 275-310 (2009).
28. A. Sopasakis & M. A. Katsoulakis, Stochastic modeling and simulation of traffic flow: ASEP with Arrhenius look-ahead dynamics, SIAM J. Appl. Math., 66, 921-944, (2006).
29. T. M. Davis, T. O.Drews, H. Ramanan, C. He, J. Dong, H. Schnablegger, M. A. Katsoulakis, E. Kokkoli, A. V. McCormick, R. L.Penn, and M. Tsapatsis. Mechanistic Principles of Nanoparticle Evolution to Zeolite Crystals, Nature Materials, 5, 400, (2006)
30. A. Chaterjee, M. A. Katsoulakis & D. G. Vlachos, Binomial distribution based τ -leap accelerated stochastic simulation, J. Chem. Phys. 122, 024112 (2005)
31. M. A. Katsoulakis, A. J. Majda, D. G. Vlachos. Coarse-grained stochastic processes for lattice systems, Proc. Natl. Acad. Sci. USA 100, 782-787, (2003).
32. M. A. Katsoulakis & A.E. Tzavaras, Contractive Relaxation Systems and the Scalar Multidimensional Conservation Law, Comm. P.D.E. 22, (1997), 195-233.
33. M. A. Katsoulakis & P. E. Souganidis, Generalized motion by mean curvature as a macroscopic limit of stochastic Ising models with long range interactions and Glauber dynamics, Comm. Math. Phys., 169, (1995), 61-97.

For a complete list see:  Google Scholar page & Researchgate profile