Deep learning methods for partial differential equations and related parameter identification problems
Recent years have witnessed a growth in mathematics for deep learning—which seeks a
deeper understanding of the concepts of deep learning with mathematics and explores how …
deeper understanding of the concepts of deep learning with mathematics and explores how …
[HTML][HTML] Learning mesh motion techniques with application to fluid–structure interaction
Mesh degeneration is a bottleneck for fluid–structure interaction (FSI) simulations and for
shape optimization via the method of mappings. In both cases, an appropriate mesh motion …
shape optimization via the method of mappings. In both cases, an appropriate mesh motion …
Bi-level iterative regularization for inverse problems in nonlinear PDEs
TTN Nguyen - Inverse Problems, 2024 - iopscience.iop.org
We investigate the ill-posed inverse problem of recovering unknown spatially dependent
parameters in nonlinear evolution partial differential equations (PDEs). We propose a bi …
parameters in nonlinear evolution partial differential equations (PDEs). We propose a bi …
A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives
G Dong, M Hintermüller, K Papafitsoros - SIAM Journal on Optimization, 2024 - SIAM
We propose and analyze a numerical algorithm for solving a class of optimal control
problems for learning-informed semilinear partial differential equations (PDEs). Such PDEs …
problems for learning-informed semilinear partial differential equations (PDEs). Such PDEs …
Sequential bi-level regularized inversion with application to hidden reaction law discovery
TTN Nguyen - arXiv preprint arXiv:2409.03834, 2024 - arxiv.org
In this article, we develop and present a novel regularization scheme for ill-posed inverse
problems governed by nonlinear partial differential equations (PDEs). In [43], the author …
problems governed by nonlinear partial differential equations (PDEs). In [43], the author …
Discretization of parameter identification in PDEs using neural networks
B Kaltenbacher, TTN Nguyen - Inverse Problems, 2022 - iopscience.iop.org
We consider the ill-posed inverse problem of identifying a nonlinearity in a time-dependent
partial differential equation model. The nonlinearity is approximated by a neural network …
partial differential equation model. The nonlinearity is approximated by a neural network …
On uniqueness in structured model learning
This paper addresses the problem of uniqueness in learning physical laws for systems of
partial differential equations (PDEs). Contrary to most existing approaches, it considers a …
partial differential equations (PDEs). Contrary to most existing approaches, it considers a …
On the identification and optimization of nonsmooth superposition operators in semilinear elliptic PDEs
C Christof, J Kowalczyk - ESAIM: Control, Optimisation and …, 2024 - esaim-cocv.org
We study an infinite-dimensional optimization problem that aims to identify the Nemytskii
operator in the nonlinear part of a prototypical semilinear elliptic partial differential equation …
operator in the nonlinear part of a prototypical semilinear elliptic partial differential equation …
CONFIDE: Contextual Finite Difference Modelling of PDEs
We introduce a method for inferring an explicit PDE from a data sample generated by
previously unseen dynamics, based on a learned context. The training phase integrates …
previously unseen dynamics, based on a learned context. The training phase integrates …
On the Identification and Optimization of Nonsmooth Superposition Operators in Semilinear Elliptic PDEs
C Christof, J Kowalczyk - arXiv preprint arXiv:2306.05185, 2023 - arxiv.org
We study an infinite-dimensional optimization problem that aims to identify the Nemytskii
operator in the nonlinear part of a prototypical semilinear elliptic partial differential equation …
operator in the nonlinear part of a prototypical semilinear elliptic partial differential equation …