Machine Learning for Data-Integrated Simulations

Computer simulation and data science have emerged as the third and fourth pillars of scientific inquiry, complementing the traditional pillars of theory and experimentation that have guided scientific progress for centuries. Data-integrated simulation, which unifies physical modeling with data-driven methods, plays an increasingly important role in modern scientific and engineering applications. This course introduces the mathematical foundations, advanced methodologies, and current research directions of data-integrated simulation with machine learning. It emphasizes the principled integration of data, physical constraints, and probabilistic modeling, and covers modern techniques including deep learning, physics-informed neural networks, and conditional generative models for uncertainty-aware simulation. Specific course contents include:

• Deep learning foundations for scientific computing: core neural network architectures, training and evaluation strategies, regularization and optimization, and practical implementation for scientific applications.
• Advanced probability and statistics: random variables and random fields, Bayesian inference, stochastic processes and stochastic differential equations, probabilistic interpretation of learning and uncertainty quantification.
• Physics-informed neural networks (PINNs): formulation and architectures of PINNs, incorporation of physical constraints, comparison with classical numerical methods and purely data-driven neural networks.
• Conditional generative models: probabilistic generative modeling for inverse problems and simulation, mathematical foundations and implementation of conditional diffusion models, and applications to uncertainty-aware data-integrated simulation.
• Research seminar and applications: student-led presentations of recent research papers in data-integrated simulation, PINNs, and generative models; critical discussion of methods, assumptions, and limitations.

• Instructor’s lectures notes, slides and assigned reading materials.

Paper or books:

• Raissi, Maziar, Paris Perdikaris, and George E. Karniadakis. "Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations." Journal of Computational physics378 (2019): 686-707.
• Ho, Jonathan, Ajay Jain, and Pieter Abbeel. "Denoising diffusion probabilistic models." Advances in neural information processing systems 33 (2020): 6840-6851.
• Song, Yang, et al. "Score-based generative modeling through stochastic differential equations." arXiv preprint arXiv:2011.13456 (2020).
• Prince, Simon JD. Understanding deep learning. MIT press, 2023.
• Goodfellow, Ian, et al. Deep learning. Vol. 1. No. 2. Cambridge: MIT press, 2016.

Oral examination lasting 40 minutes.

Before the examination, consultation sessions will be offered and announced in due time via Ilias or during the lecture. These sessions are primarily intended to clarify any remaining questions that arise during exam preparation.

Additionally, to summarize the lecture and support exam preparation, we will offer an overview lecture on the exam content. In this session, the general structure and procedure of the exam as well as example questions will be presented. The materials for the overview lecture will be made available on Ilias at the appropriate time.