Jan Martin Nicolaus

Doktorand

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Raum:
2.29.2.05
Telefon:
+49 331 977-230182

Publikationen

2024 | Nonintrusive model order reduction for stochastic differential equations | M. A. Freitag, J. M. Nicolaus, M. RedmannZeitschrift: arXivLink zum Preprint

Nonintrusive model order reduction for stochastic differential equations

Autoren: M. A. Freitag, J. M. Nicolaus, M. Redmann (2024)

A non-intrusive model order reduction method for bilinear stochastic differential equations with additive noise is proposed. A reduced order model (ROM) is designed in order to approximate the statistical properties of high-dimensional systems. The drift and diffusion coefficients of the ROM are inferred from state observations by solving appropriate least-squares problems. The closeness of the ROM obtained by the presented approach to the intrusive ROM obtained by the proper orthogonal decomposition (POD) method is investigated. Two generalisations of the snapshot-based dominant subspace construction to the stochastic case are presented. Numerical experiments are provided to compare the developed approach to POD.

Zeitschrift:
arXiv

2023 | Model order reduction methods applied to neural network training | M.A. Freitag, J.M. Nicolaus, M. RedmannZeitschrift: Proceedings in Applied Mathematics and MechanicsSeiten: e202300078Link zur Publikation

Model order reduction methods applied to neural network training

Autoren: M.A. Freitag, J.M. Nicolaus, M. Redmann (2023)

Neural networks have emerged as powerful and versatile tools in the field of deep learning. As the complexity of the task increases, so do size and architectural complexity of the causing compression techniques to become a focus of current research. Parameter truncation can provide a significant reduction in memory and computational complexity. Originating from a model order reduction framework, the Discrete Empirical Interpolation Method is applied to the gradient descent training of neural networks and analyze for important parameters. The approach for various state-of-the-art neural networks is compared to established truncation methods. Further metrics like L2 and Cross-Entropy Loss, as well as accuracy and compression rate are reported.

Zeitschrift:
Proceedings in Applied Mathematics and Mechanics
Seiten:
e202300078

2023 | Can one hear the depth of the water? | M.A. Freitag, P.Kriz, T. Mach, J. M. NicolausZeitschrift: Proceedings in Applied Mathematics and MechanicsSeiten: e202300122Link zur Publikation

Can one hear the depth of the water?

Autoren: M.A. Freitag, P.Kriz, T. Mach, J. M. Nicolaus (2023)

We discuss discrete-time dynamical systems depending on a parameter μ. Assuming that the system matrix A(μ) is given, but the parameter μ is unknown, we infer the most-likely parameter μm≈μ from an observed trajectory x of the dynamical system. We use parametric eigenpairs (vi(μ),lambdai(μ) of the system matrix A(μ) computed with Newton's method based on a Chebyshev expansion. We then represent x in the eigenvector basis defined by the vi(μ) and compare the decay of the components with predictions based on the lambdai(μ). The resulting estimates for μ are combined using a kernel density estimator to find the most likely value for μm and a corresponding uncertainty quantification.

Zeitschrift:
Proceedings in Applied Mathematics and Mechanics
Seiten:
e202300122