Data-Driven Mathematical Modeling

The group contributes primarily to the following mathematical research topics of WIAS:

Hysteresis operators and rate-independent systems

Time-dependent processes in physics, biology, and economics often exhibit a rate-independent input-output behavior. Quite often, such processes are accompanied by the occurrence of hysteresis phenomena induced by inherent memory effects. There are two methods to describe such processes at WIAS: rate independent systems and. hysteresis operators . [>> more]

Machine Learning: Mathematical foundations to applications

Machine learning has become one of the driving forces of modern science and technology. As data grows in volume and complexity, it offers powerful tools to uncover hidden structures, make predictions, and support decisions in situations where traditional modeling reaches its limits. At WIAS, machine learning is approached from a mathematical perspective and applied to various areas. Investigations are focused on making machine learning more efficient, reliable, and interpretable. In that line, methods that combine data-driven models with physical principles are developed. [>> more]

Multi-scale modeling, asymptotic analysis, and hybrid models

To understand the interplay between different physical effects one often needs to consider models involving several length scales. The aim in this mathematical topic is the derivation of effective models for the efficient description of the processes. The understanding of the transfer between different scales relies on mathematical methods such as homogenization, asymptotic analysis, or Gamma convergence. The generated effective models are coupled partial differential equations combining volume and interfacial effects. [>> more]

Optimal control of partial differential equations and nonlinear optimization

Many processes in nature and technics can only be prescribed by partial differential equations,e.g. heating- or cooling processes, the propagation of acoustic or electromagnetic waves, fluid mechanics. Additionally to challenges in modeling, in various applications the manipulation or controlling of the modeled system is also of interest in order to obtain a certain purpose... [>> more]

Stochastic Optimization

Stochastic Optimization in the widest sense is concerned with optimization problems influenced by random parameters in the objective or constraints. [>> more]

Systems of partial differential equations: modeling, numerical analysis and simulation

The mathematical modelling of many scientific and technological problems leads to (initial) boundary value problems with systems of partial differential equations (PDEs). [>> more]

Archive

Further mathematical research topics where the institute has expertise in:

Direct and inverse problems for the Maxwell equations

The work is focussed on models for inductive heating of steel and for light scattering by periodic surface structures. For this the quasi-stationary Maxwell equation is coupled with nonlinear partial differential equations and the timeharmonic Maxwell equation is combined with special radiation conditions, respectively. The convergence of numerical methods and several inverse promblems are analyzed. [>> more]

Magnetohydrodynamics

For the production of semiconductor crystals, electromagnetic fields are often used to produce heat by induction. Moreover, Lorentz forces can improve the melt motion during crystal growth processes. Their modeling leads to a system of coupled partial differential equations. [>> more]

Nonlinear kinetic equations

Kinetic equations describe the rate at which a system or mixture changes its chemical properties. Such equations are often non-linear, because interactions in the material are complex and the speed of change is dependent on the system size as well as the strength of the external influences. [>> more]

Plates, Beams, Shells and Arches

An efficient description of the mechanical behavior of special 3D bodies, whose dimensions in one or two directions are small in comparison to the other dimensions, is possible with so-called plate models or beam models. [>> more]