FakET: Simulating Cryo-Electron Tomograms with Neural Style Transfer

Particle localization and -classification constitute two of the most fundamental problems in computational microscopy. In recent years, deep learning based approaches have been introduced for these tasks with great success. A key shortcoming of these supervised learning methods is their need for large training data sets, typically generated from particle models in conjunction with complex numerical forward models simulating the physics of transmission electron microscopes. Computer implementations of such forward models are computationally extremely demanding and limit the scope of their applicability. In this paper we propose a simple method for simulating the forward operator of an electron microscope based on additive noise and Neural Style Transfer techniques. We evaluate the method on localization and classification tasks using one of the established state-of-the-art architectures showing performance on par with the benchmark. In contrast to previous approaches, our method accelerates the data generation process by a factor of 750 while using 33 times less memory and scales well to typical transmission electron microscope detector sizes. It utilizes GPU acceleration and parallel processing. It can be used as a stand-alone method to adapt a training data set or as a data augmentation technique. The source code is available at https://gitlab.com/deepet/faket.

Neural and gpc operator surrogates: construction and expression rate bounds

Approximation rates are analyzed for deep surrogates of maps between infinite-dimensional function spaces, arising e.g. as data-to-solution maps of linear and nonlinear partial differential equations. Specifically, we study approximation rates for Deep Neural Operator and Generalized Polynomial Chaos (gpc) Operator surrogates for nonlinear, holomorphic maps between infinite-dimensional, separable Hilbert spaces. Operator in- and outputs from function spaces are assumed to be parametrized by stable, affine representation systems. Admissible representation systems comprise orthonormal bases, Riesz bases or suitable tight frames of the spaces under consideration. Algebraic expression rate bounds are established for both, deep neural and gpc operator surrogates acting in scales of separable Hilbert spaces containing domain and range of the map to be expressed, with finite Sobolev or Besov regularity. We illustrate the abstract concepts by expression rate bounds for the coefficient-to-solution map for a linear elliptic PDE on the torus.

Deep neural network approximation for high-dimensional parabolic Hamilton-Jacobi-Bellman equations

The approximation of solutions to second order Hamilton--Jacobi--Bellman (HJB) equations by deep neural networks is investigated. It is shown that for HJB equations that arise in the context of the optimal control of certain Markov processes the solution can be approximated by deep neural networks without incurring the curse of dimension. The dynamics is assumed to depend affinely on the controls and the cost depends quadratically on the controls. The admissible controls take values in a bounded set.

Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions

In recent work it has been established that deep neural networks are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been restricted to problems formulated on the whole Euclidean domain. On the other hand, most problems in engineering and the sciences are formulated on finite domains and subjected to boundary conditions. The present paper considers an important such model problem, namely the Poisson equation on a domain $D\subset \mathbb{R}^d$ subject to Dirichlet boundary conditions. It is shown that deep neural networks are capable of representing solutions of that problem without incurring the curse of dimension. The proofs are based on a probabilistic representation of the solution to the Poisson equation as well as a suitable sampling method.