课程一   

• 课程名称：Mathematical Theory for Deep Neural Networks

• 授课老师：许进超（King Abdullah University of Science & Technology）

• 授课时间：2026/7/6-2026/7/10

•  

• 教学内容：

• This course provides a rigorous introduction to the mathematical foundations of neural network functions and their applications in numerical analysis. Students will explore the deep connections between finite element methods and neural network architectures, universal approximation theories,and modern training algorithms. The course emphasizes theoretical concepts such as Barron spaces,ReLUk networks, and their application to solving partial differential equations.

•  

• 教学大纲：

• 1. An introduction to finite element spaces

  2. Neural networks as a generalization of finite element functions

  3. Generating arbitrary h-p finite element spaces by ReLU–ReLUk -DNNs

  4. Universal approximation theorem

  5. Concentration inequalities and a simple analysis of cosine neural networks

  6. Barron spaces, representation theorem, and basic error estimates

  7. Approximation theory for ReLUk neural networks

  8. Training algorithms and the frequency principle

  9. Greedy algorithms

  10. Linearized shallow neural networks

  11. Finite neuron methods

  12. Metric entropy: curse and no-curse of dimensionality

课程二

• 课程名称：Computational Multiscale Methods for Partial Differential Equations

• 授课老师：Daniel Peterseim (University of Augsburg)

• 授课时间：2026/7/6-2026/7/17

•  

• 教学内容：

• Many physical and engineering systems are governed by partial differential equations (PDEs) whose coefficients or solutions exhibit features across multiple spatial scales. Direct numerical simulation of such problems is often computationally infeasible due to the high resolution required to capture fine-scale effects. Computational multiscale methods address this challenge by incorporating fine-scale information into coarse-scale numerical schemes without explicitly resolving all scales.

   

  This course is centered on numerical homogenization via localized orthogonal decomposition (LOD), which is developed as a concrete and mathematically well-founded approach to multiscale discretization of elliptic PDEs with rough coefficients. The course covers the analytical foundations of numerical homogenization, the decomposition of scales, localization phenomena, and the algorithmic realization of LOD-based methods in detail. Applications of LOD techniques to nonlinear eigenvalue problems are presented to illustrate their practical impact. In addition, selected discussions of learning-based and hybrid classical–quantum algorithmic extensions are included. An optional concluding lecture provides an outlook on variational relaxation and microstructure formation in nonlinear elasticity, which lies beyond the LOD framework.

•  

• 教学大纲：

• 1. Multiscale Problems and Numerical Homogenization

  Multiscale phenomena in PDEs; oscillatory coefficients; pre-asymptotic effects; limitations of classical discretizations.

  2. Elliptic Homogenization from a Numerical Perspective

  Oscillatory diffusion problems; effective coefficients; periodic homogenization; numerical homogenization beyond scale separation.

  3. Decomposition of Scales and Ideal Numerical Homogenization

  Finite element spaces; quasi-interpolation; orthogonal decomposition of scales; ideal numerical homogenization.

  4. Localization of Numerical Correctors

  Exponential decay of correctors; localization strategies; domain decomposition; connections to eigenvector localization.

  5. Localized Orthogonal Decomposition (LOD)

  Standard LOD formulation; Petrov–Galerkin variants; boundary conditions; high-contrast diffusion problems.

  6. Error Analysis and Computational Aspects of LOD-Based Methods

  A priori error estimates; localization parameters; computational complexity; implementation aspects.

  7. Effective Coefficients and Relation to Periodic Homogenization

  Quasi-local and local effective coefficients; connections to the mathematical theory of homogenization.

  8. Applications: Nonlinear Eigenvalue Problems

  Localization phenomena in multiscale eigenvalue problems; spectral effects; metric-based viewpoints.

  9. Algorithmic Extensions and Outlook

  Learning-based acceleration of multiscale correctors; operator learning perspectives; hybrid classical–quantum approaches.

  10. Beyond LOD-Type Methods: Relaxation in Nonlinear Elasticity (optional)

  Nonconvex variational integrals; microstructure formation; relaxation by (semi)convexification; polyconvexity and rank-one convex hulls.

•  

  Selected References

  [1] R. Altmann, P. Henning, and D. Peterseim. Quantitative anderson localization of schr¨odinger eigenstates under disorder potentials. Mathematical Models and Methods in Applied Sciences, 30(05):917–955, 2020.

  [2] R. Altmann, P. Henning, and D. Peterseim. Numerical homogenization beyond scale separation. Acta Numer., 30:1–86, 2021.

  [3] M. Deiml and D. Peterseim. Quantum Realization of the Finite Element Method. Math.Comp., 2025.

  [4] D. Gallistl and D. Peterseim. Computation of quasi-local effective diffusion tensors and connections to the mathematical theory of homogenization. SIAM Multiscale Model. Simul., 15(4), 2017.

  [5] M. Hauck and D. Peterseim. Super-localization of elliptic multiscale problems. Math.Comp., 92(342):981–1003, 2022.

  [6] P. Henning, L. Huynh, and D. Peterseim. Metric-driven numerical methods. arXiv e-print, 2512.10083, 2025.

  [7] R. Kornhuber, D. Peterseim, and H. Yserentant. An analysis of a class of variational multiscale methods based on subspace decomposition. Math. Comp., 87(314):2765–2774, 2018.

  [8] F. Kr¨opfl, R. Maier, and D. Peterseim. Operator compression with deep neural networks. Adv. Contin. Disc. Model., 2022-29, 2022.

  [9] A. M˚alqvist and D. Peterseim. Numerical homogenization by localized orthogonal decomposition, volume 5 of SIAM Spotlights. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, [2021] ©2021.

  [10] T. Neumeier, M. A. Peter, D. Peterseim, and D. Wiedemann. Computational polyconvexification of isotropic functions. Multiscale Modeling & Simulation, 22(4):1402–1420, 2024.

课程三​

• 课程名称：Heaviside Composite Optimization: A new paradigm in optimization (10 lectures)

• 授课老师：Jong-Shi Pang（University of Southern California）

• 授课时间：2026/7/20-2026/7/24 

•  

• 教学内容：

• This course introduces the topic of Heaviside composite optimization and covers its many facets: breadth in modeling, roles in old and new applications, theory of optimizers and stationary solutions, bridge with discrete optimization, and the progressive integer programming method. By definition, a univariate Heavisidefunction is the (discontinuous) indicator of an interval. By its name, a Heaviside composite function is the composition of a Heaviside function with a continuous multivariate function that maybe nonconvex and nondifferentiable. While very natural in modeling many physical phenomena involving paradigm switches, a Heaviside composite optimization problem, possibly with Heaviside composite functional constraints, has never been formally studied. Our work aims to fill this void with a comprehensive research program covering the applications, theory, and algorithms for this novel class of very challenging optimization problems.

• 教学大纲：

• This course is based on a monograph in progress that is co-authored with Junyi Liu. The lectures are divided into 5 parts:

• 1. Introduction and sources: paradigm switches and discontinuities

• 2. Problems with affine combinations: theory and algorithms

• 3. Extended problems: complementarity constraints, fractional programs, and double composition

• 4. Modeling, computational implementations, and numerical results (to be delivered by Junyi Liu)

• 5. Conditional optimization in statistics and policy learning.

•  

课程四

• 课程名称：Noncooperative Games and Extensions: From classical to modern (10 lectures)

• 授课老师：Jong-Shi Pang（University of Southern California）

• 授课时间：2026/7/27-2026/7/31

•  

• 教学内容：

• Mathematical game theory provides a rigorous approach for the resolution of conflicts among uncoordinated entities. Noncooperative games (or simply, games) represent an important sub-field of game theory and find relevance in modeling a multitude of decision-making problems in the presence of selfish decision makers, rival competition, system disruptions, and adversarial attacks, all with or without event uncertainties and account of risks. Having originated from economics, the study and applications of noncooperative game theory have spread well beyond the pioneering of von Neumann, Morgernstern, Arrow and Debreu to name a few of the pioneers of the field.  Some modern applied areas in engineering include the pricing and distribution of electricity in energy markets, spectrum management in signal processing and communication networks, adversarial degrading of system performance, disruptions of deep neural networks in machine learning, and most recently, traffic equilibrium with the e-sharing travel mode (besides solo driving) and mixed-autonomy systems that include autonomous and electric vehicles (in addition to human drivers).

•  

• 教学大纲：

• This course is based on a monograph in progress that is co-authored with Uday Shanbhag who has contributed significantly to game under uncertainty, a topic that we omit in these lectures. The 10 lectures here are divided into 5 parts:

• 1. Introduction and some sources: Classes of games and solution concepts

• 2. Single-level games: Solution analysis

• 3. Multi-leader-follower games: analysis

• 4. Computational algorithms

• 5. Recent game models in traffic and fusion with electricity markets

课程五

• 课程名称：Computational Gradient Flows and Optimal Transport Theory and Applications in Machine Learning and Statistics

• 授课老师：Jia-Jie Zhu（KTH Royal Institute of Technology）

• 授课时间：2026/7/20-2026/7/31

•  

• 教学内容：

• This course offers a self-contained, first-principles introduction to gradient flows of probability measures and optimal transport, aimed at graduate students and researchers in computational mathematics, machine learning, statistics, as well as other engineering and scientific disciplines. Building on the theoretical foundation of optimal transport and PDE, our focus is a principled view of the Wasserstein gradient structure and its computational aspects especially in the context of machine learning and statistics. No prior background in PDE or stochastic analysis is assumed; we will try to be self-contained for a wide audience interested in both theory and computation.

•  

• 教学大纲：

• 1. Introduction to gradient flows in the Banach and Hilbert space

  • From Euclidean gradient descent to Wasserstein gradient flow

  • PDE and gradient structure

  2. Introduction to optimal transport of probability distributions

  • The Wasserstein distance

  • The Fisher–Rao distance

  • Unbalanced optimal transport: entropy relaxation and lifting construction

  3. Gradient flows of probability measures

  • Wasserstein gradient flows

  • Fisher–Rao gradient flow and natural gradient

  • Unbalanced transport and Hellinger–Kantorovich gradient flows

  4. Approximation of gradient flows

  • Interaction particle systems, blob method, Stein gradient flows

  • Reduced gradient structure, Gaussian gradient flows

  5. Applications

  • Statistical inference, sampling

  • Machine learning

  • Control and optimization