Priority Program (SPP 2298)

Theoretical Foundations of Deep Learning


Project Hu1889/7-1: Deep neural networks overcome the curse of dimensionality in the numerical approximation of stochastic control problems and of semilinear Poisson equations - Martin Hutzenthaler and Thomas Kruse (Giessen) (2022-2025)

Partial differential equations (PDEs) are a key tool in the modeling of many real world phenomena. Several PDEs that arise in financial engineering, economics, quantum mechanics or statistical physics are nonlinear, high-dimensional, and cannot be solved explicitly. It is a highly challenging task to provably solve such high-dimensional nonlinear PDEs approximately without suffering from the so-called curse of dimensionality. Deep neural networks (DNNs) and other deep learning-based methods have recently been applied very successfully to a number of computational problems. In particular, simulations indicate that algorithms based on DNNs overcome the curse of dimensionality in the numerical approximation of solutions of certain nonlinear PDEs. For certain linear and nonlinear PDEs this has also been proven mathematically. The key goal of this project is to rigorously prove for the first time that DNNs overcome the curse of dimensionality for a class of nonlinear PDEs arising from stochastic control problems and for a class of semilinear Poisson equations with Dirichlet boundary conditions.  





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Research grant

DFG Project Hu1889/6-1 and Hu1889/6-2: On numerical approximations of high-dimensional nonlinear parabolic partial differential equations and of backward stochastic differential equations - Martin Hutzenthaler (2017-2023)

Parabolic partial differential equations (PDEs) are a fundamental tool in the state-of-the-art pricing and hedging of financial derivatives. The PDEs appearing in such financial engineering applications are often high- dimensional and nonlinear. Since explicit solutions of such PDEs are typically not available, it is a very active topic of research to solve such PDEs approximately. However existing approximation methods are either computationally expensive or are only applicable to a (in our view) small class of PDEs. The goal of this project is to establish a class of generally applicable numerical approximations for semilinear parabolic partial differential equations and, in particular, to prove that the computational effort grows at most linearly in the dimension and cubic in the reciprocal of the prescribed accuracy. In particular, in the notation of information-based complexity, our goal is to show that the numerical problem of approximating the solution of a semilinear parabolic PDE at single space-time points is polynomially tractable.



SPP 1590

Priority Program (SPP 1590)

Probabilistic Structures in Evolution


Project Hu1889/3-1 and Hu1889/3-2: Evolution of altruistic defense traits in structured population - Martin Hutzenthaler and Dirk Metzler (Munich) (2011-2020)

In the first project phase we investigated under which conditions an inheritable behavioral trait of defense against parasites can spread in a structured population even if it is costly in the sense that individuals having a defense gene tend to have less offspring. In this proposed continuation project we study in a many-demes limit the time until the first fixation of a defense allele arising from rare mutations. We are going to show that this time to first fixation is logarithmic in the inverse mutation rate. So even for small mutation rates defense traits can appear on an evolutionary relevant time scale. Mathematically our central contribution is to prove and generalize the results of Dawson and Greven (2011) without using dual processes for a large class of processes. 




Project Hu1889/4-1: The effect of natural selection on genealogies - Martin Hutzenthaler and Peter Pfaffelhuber (Freiburg) (2015-2020)

Natural selection shapes genealogies within a population in various ways. In our proposal, we suggest both, a general qualitative study of some aspects of genealogies under selection, and a quantitative treatment of specific relevant models. More precisely, we study models with unbounded selection (so there are arbitraryly beneficial and/or deleterious fitness classes) using Girsanov transforms and approximate dualities, and selection in fluctuating environment (where an allele can be beneficial or deleterious, depending on the environment) using a general result on stochastic averaging in the limit of fast environmental changes. For both models, we use the previously developed technique of treating genealogical trees as metric measure spaces, leading to tree-valued stochastic Markov processes. The qualitative work is dealing with a comparison of genealogical distances under neutrality and under selection. We conjecture that many ssituations including selection lead to shorter genealogical distances.



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GRK 2131

Research training school (RTG) 2131 (2014-2020)

High-dimensional Phenomena in Probability -
Fluctuations and Discontinuity


The Research Training Group (RTG) High-dimensional Phenomena in Probability - Fluctuations and Discontinuity offers excellent national and international graduates in the mathematical sciences the opportunity to conduct internationally visible doctoral research in probability theory. The goal of the RTG is to bring together the joint expertise on aspects of high dimension in probability. In the study of random structures in high dimensions, one frequently observes universality in limit theorems (fluctuations) as well as phase transitions (discontinuities). These aspects form the common focus of a large number of currently active research projects in stochastic processes. The cooperation of several research groups will offer the Ph.D. students the unique opportunity to gain experience beyond their own research topic, thus giving a broad scientific education. The RTG is supported by top level research groups in probability theory and its applications, stochastic analysis, stochastic geometry and mathematical physics. The research groups involved in the RTG have recently successfully carried out externally funded research projects in probability and statistics. As a rule, each doctoral student in the RTG will be supervised by two PIs.



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Research grant

DFG Project Hu1889/2-1: Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients - Martin Hutzenthaler (2012-2014)

Stochastic differential equations (SDEs) are used in all areas for modeling dynamics with stochastic noise. As applied SDEs typically admit no explicit solution, it is crucial to solve SDEs numerically. The majority of applied SDEs have superlinearly growing coefficients and, therefore, do not satisfy the assumptions of the bulk of the literature. We have recently shown that algorithms developed for the case of global Lipschitz coefficients do in general not transfer to the non-global Lipschitz case without modifications. For this reason, we investigate the convergence behavior of suitably modified explicit Euler methods. More precisely we develop a thorough theory of numerical methods which are recursively defined as a general function of the previous state, of the time increment and of the increment of the Brownian motion. The convergence theory will apply to most of the stochastic ordinary differential equations with locally Lipschitz continuous coefficients having finite moments. Establishing the order of convergence will require additional assumptions such as local smoothness. Our main approach is to bring forward the successful Lyapunov technique to the theory of numerical approximations. Moreover, we extend our finite-dimensional results to stochastic partial differential equations. In particular we study a modified version of the exponential Euler method which hasrecently been proposed for the case of additive noise and which has a rather good order of convergence.



FOR 1078

Project Hu1889/1-1: Competing selective sweeps - Martin Hutzenthaler and Peter Pfaffelhuber (Freiburg) (2011-2015)

The fixation of a positively selected allele reduces linked neutral sequence diversity, an effect known as a “selective sweep”. The literature on recurrent selective sweeps has so far mainly focused on the non-overlapping case where at most a single beneficial allele sweeps to fixation at any time. In this project, we study competing selective sweeps where further beneficial mutations arise at different recombining loci during the time-course of the first selective sweep. For example, such a scenario is realistic for a structured population with low migration rates or with previously isolated demes (subpopulations). Recombination events can then bring beneficial alleles to the same genetic background. If such recombination events happen more than once, we expect that competing selective sweeps leave the following distinct genetic footprint: Between selected loci, there is a strong haplotype structure; outside the selected loci, there is a severe reduction of genetic diversity, similar to the case of a single selective sweep. We analyze and apply this model of competing selective sweeps in three steps: (1) The genealogy at a neutral locus, linked to the beneficial alleles, can be studied using the ancestral selection graph. Using this genealogical picture we will quantify the genetic footprint. (2) Using simulation techniques based on the recently developed software MSMS and its proposed extension, we establish and analyze the distinct genetic footprint of competing sweeps in panmictic and structured populations. In particular, we explore possible sources of a strong haplotype structure. (3) In collaboration with empirical groups of the research unit, we scan for the signature of competing sweeps in SNP data of Drosophila melanogaster and in SNP data of Solanum chilense and Solanum peruvianum.



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