9 papers
Tensor-Train Compressed Separable PINNs: A Curvature-Aware Optimization Framework for Parametric PDEs in High Dimensions
Denis Korolev, Martin Eigel
In this work, we develop a second-order optimization framework for physics-informed neural networks (PINNs) applied to high-dimensional parametric partial differential equations (P…
Multilevel Sparse Tensor Approximation for High-Dimensional Parametric PDEs
Martin Eigel, Philipp Trunschke, Dana Wrischnig
In this paper the efficiency of multilevel sparse tensor approximation methods for high-dimensional affine parametric diffusion equations is investigated. Methodologically, the rec…
Learning Transient Convective Heat Transfer with Geometry Aware World Models
Onur T. Doganay, Alexander Klawonn, Martin Eigel +1
Partial differential equation (PDE) simulations are fundamental to engineering and physics but are often computationally prohibitive for real-time applications. While generative AI…
Approximation and learning with compositional tensor trains
Martin Eigel, Charles Miranda, Anthony Nouy +1
We introduce compositional tensor trains (CTTs) for the approximation of multivariate functions, a class of models obtained by composing low-rank functions in the tensor-train form…
Multi-level Neural Networks for high-dimensional parametric obstacle problems
Martin Eigel, Cosmas Heiß, Janina E. Schütte
A new method to solve computationally challenging (random) parametric obstacle problems is developed and analyzed, where the parameters can influence the related partial differenti…
Sampling from Boltzmann densities with physics informed low-rank formats
Paul Hagemann, Janina Schütte, David Sommer +2
Our method proposes the efficient generation of samples from an unnormalized Boltzmann density by solving the underlying continuity equation in the low-rank tensor train (TT) forma…
Multilevel CNNs for Parametric PDEs based on Adaptive Finite Elements
Janina Enrica Schütte, Martin Eigel
A neural network architecture is presented that exploits the multilevel properties of high-dimensional parameter-dependent partial differential equations, enabling an efficient app…
Adaptive Multilevel Neural Networks for Parametric PDEs with Error Estimation
Janina E. Schütte, Martin Eigel
To solve high-dimensional parameter-dependent partial differential equations (pPDEs), a neural network architecture is presented. It is constructed to map parameters of the model d…
Functional SDE approximation inspired by a deep operator network architecture
Martin Eigel, Charles Miranda
A novel approach to approximate solutions of Stochastic Differential Equations (SDEs) by Deep Neural Networks is derived and analysed. The architecture is inspired by the notion of…