401465621L Deep Learning in Scientific Computing
Semester  Spring Semester 2021 
Lecturers  S. Mishra 
Periodicity  nonrecurring course 
Language of instruction  English 
Comment  Aimed at students in a Master's Programme in Mathematics, Engineering and Physics. 
Courses
Number  Title  Hours  Lecturers  

401465621 V  Deep Learning in Scientific Computing  2 hrs 
 S. Mishra  
401465621 U  Deep Learning in Scientific Computing  1 hrs 
 S. Mishra 
Catalogue data
Abstract  Machine Learning, particularly deep learning is being increasingly applied to perform, enhance and accelerate computer simulations of models in science and engineering. This course aims to present a highly topical selection of themes in the general area of deep learning in scientific computing, with an emphasis on the application of deep learning algorithms for systems, modeled by PDEs. 
Objective  The objective of this course will be to introduce students to advanced applications of deep learning in scientific computing. The focus will be on the design and implementation of algorithms as well as on the underlying theory that guarantees reliability of the algorithms. We will provide several examples of applications in science and engineering where deep learning based algorithms outperform state of the art methods. 
Content  A selection of the following topics will be presented in the lectures. 1. Issues with traditional methods for scientific computing such as Finite Element, Finite Volume etc, particularly for PDE models with highdimensional state and parameter spaces. 2. Introduction to Deep Learning: Artificial Neural networks, Supervised learning, Stochastic gradient descent algorithms for training, different architectures: Convolutional Neural Networks, Recurrent Neural Networks, ResNets. 3. Theoretical Foundations: Universal approximation properties of the Neural networks, BiasVariance decomposition, Bounds on approximation and generalization errors. 4. Supervised deep learning for solutions fields and observables of highdimensional parametric PDEs. Use of lowdiscrepancy sequences and multilevel training to reduce generalization error. 5. Uncertainty Quantification for PDEs with supervised learning algorithms. 6. Deep Neural Networks as Reduced order models and prediction of solution fields. 7. Active Learning algorithms for PDE constrained optimization. 8. Recurrent Neural Networks and prediction of time series for dynamical systems. 9. Physics Informed Neural networks (PINNs) for the forward problem for PDEs. Applications to highdimensional PDEs. 10. PINNs for inverse problems for PDEs, parameter identification, optimal control and data assimilation. All the algorithms will be illustrated on a variety of PDEs: diffusion models, BlackScholes type PDEs from finance, wave equations, Euler and NavierStokes equations, hyperbolic systems of conservation laws, Dispersive PDEs among others. 
Lecture notes  Lecture notes will be provided at the end of the course. 
Literature  All the material in the course is based on research articles written in last 12 years. The relevant references will be provided. 
Prerequisites / Notice  The students should be familiar with numerical methods for PDEs, for instance in courses such as Numerical Methods for PDEs for CSE, Numerical analysis of Elliptic and Parabolic PDEs, Numerical methods for hyperbolic PDEs, Computational methods for Engineering Applications. Some familiarity with basic concepts in machine learning will be beneficial. The exercises in the course rely on standard machine learning frameworks such as KERAS, TENSORFLOW or PYTORCH. So, competence in Python is helpful. 
Performance assessment
Performance assessment information (valid until the course unit is held again)  
Performance assessment as a semester course  
ECTS credits  6 credits 
Examiners  S. Mishra 
Type  graded semester performance 
Language of examination  English 
Repetition  Repetition only possible after reenrolling for the course unit. 
Learning materials
No public learning materials available.  
Only public learning materials are listed. 
Groups
401465621 U  Deep Learning in Scientific Computing  
Group  G01 

Restrictions
Places  200 at the most 
Waiting list  until 01.03.2021 
Offered in
Programme  Section  Type  

Data Science Master  Core Electives  W  
Mathematics Master  Selection: Numerical Analysis  W  
Computational Science and Engineering Master  Electives  W 