arXiv subjects

Christian Lessig

Publications and source records attributed to Christian Lessig.

At least 37 records · Page 2Linked to original sources

arXiv2023

DNN-MG: A Hybrid Neural Network/Finite Element Method with Applications to 3D Simulations of the Navier-Stokes Equations

We extend and analyze the deep neural network multigrid solver (DNN-MG) for the Navier-Stokes equations in three dimensions. The idea of the method is to augment a finite element simulation on coarse grids with fine scale information obtained using deep neural networks. The neural network operates locally on small patches of grid elements. The local approach proves to be highly efficient, since the network can be kept (relatively) small and since it can be applied in parallel on all grid patches. However, the main advantage of the local approach is the inherent generalizability of the method. Since the network only processes data of small sub-areas, it never ``sees'' the global problem and thus does not learn false biases. We describe the method with a focus on the interplay between the finite element method and deep neural networks. Further, we demonstrate with numerical examples the excellent efficiency of the hybrid approach, which allows us to achieve very high accuracy with a coarse grid and thus reduce the computation time by orders of magnitude.

math.NA↗

arXiv2023

AtmoRep: A stochastic model of atmosphere dynamics using large scale representation learning

The atmosphere affects humans in a multitude of ways, from loss of life due to adverse weather effects to long-term social and economic impacts on societies. Computer simulations of atmospheric dynamics are, therefore, of great importance for the well-being of our and future generations. Here, we propose AtmoRep, a novel, task-independent stochastic computer model of atmospheric dynamics that can provide skillful results for a wide range of applications. AtmoRep uses large-scale representation learning from artificial intelligence to determine a general description of the highly complex, stochastic dynamics of the atmosphere from the best available estimate of the system's historical trajectory as constrained by observations. This is enabled by a novel self-supervised learning objective and a unique ensemble that samples from the stochastic model with a variability informed by the one in the historical record. The task-independent nature of AtmoRep enables skillful results for a diverse set of applications without specifically training for them and we demonstrate this for nowcasting, temporal interpolation, model correction, and counterfactuals. We also show that AtmoRep can be improved with additional data, for example radar observations, and that it can be extended to tasks such as downscaling. Our work establishes that large-scale neural networks can provide skillful, task-independent models of atmospheric dynamics. With this, they provide a novel means to make the large record of atmospheric observations accessible for applications and for scientific inquiry, complementing existing simulations based on first principles.

physics.ao-ph↗

arXiv2022

A Multi-Scale Deep Learning Framework for Projecting Weather Extremes

Weather extremes are a major societal and economic hazard, claiming thousands of lives and causing billions of dollars in damage every year. Under climate change, their impact and intensity are expected to worsen significantly. Unfortunately, general circulation models (GCMs), which are currently the primary tool for climate projections, cannot characterize weather extremes accurately. To address this, we present a multi-resolution deep-learning framework that, firstly, corrects a GCM's biases by matching low-order and tail statistics of its output with observations at coarse scales; and secondly, increases the level of detail of the debiased GCM output by reconstructing the finer scales as a function of the coarse scales. We use the proposed framework to generate statistically realistic realizations of the climate over Western Europe from a simple GCM corrected using observational atmospheric reanalysis. We also discuss implications for probabilistic risk assessment of natural disasters in a changing climate.

cs.LG↗

arXiv2022

AtmoDist: Self-supervised Representation Learning for Atmospheric Dynamics

Representation learning has proven to be a powerful methodology in a wide variety of machine learning applications. For atmospheric dynamics, however, it has so far not been considered, arguably due to the lack of large-scale, labeled datasets that could be used for training. In this work, we show that the difficulty is benign and introduce a self-supervised learning task that defines a categorical loss for a wide variety of unlabeled atmospheric datasets. Specifically, we train a neural network on the simple yet intricate task of predicting the temporal distance between atmospheric fields from distinct but nearby times. We demonstrate that training with this task on ERA5 reanalysis leads to internal representations capturing intrinsic aspects of atmospheric dynamics. We do so by introducing a data-driven distance metric for atmospheric states. When employed as a loss function in other machine learning applications, this Atmodist distance leads to improved results compared to the classical $\ell_2$-loss. For example, for downscaling one obtains higher resolution fields that match the true statistics more closely than previous approaches and for the interpolation of missing or occluded data the AtmoDist distance leads to results that contain more realistic fine scale features. Since it is derived from observational data, AtmoDist also provides a novel perspective on atmospheric predictability.

physics.ao-ph↗

arXiv2021

Variational symplectic diagonally implicit Runge-Kutta methods for isospectral systems

Isospectral flows appear in a variety of applications, e.g. the Toda lattice in solid state physics or in discrete models for two-dimensional hydrodynamics, with the isospectral property often corresponding to mathematically or physically important conservation laws. Their most prominent feature, i.e. the conservation of the eigenvalues of the matrix state variable, should therefore be retained when discretizing these systems. Recently, it was shown how isospectral Runge-Kutta methods can, in the Lie-Poisson case also considered in our work, be obtained through Hamiltonian reduction of symplectic Runge-Kutta methods on the cotangent bundle of a Lie group. We provide the Lagrangian analogue and, in the case of symplectic diagonal implicit Runge-Kutta methods, derive the methods through a discrete Euler-Poincare reduction. Our derivation relies on a formulation of diagonally implicit isospectral Runge-Kutta methods in terms of the Cayley transform, generalizing earlier work that showed this for the implicit midpoint rule. Our work is also a generalization of earlier variational Lie group integrators that, interestingly, appear when these are interpreted as update equations for intermediate time points. From a practical point of view, our results allow for a simple implementation of higher order isospectral methods and we demonstrate this with numerical experiments where both the isospectral property and energy are conserved to high accuracy.

math.NA↗

arXiv2021

Towards Representation Learning for Atmospheric Dynamics

The prediction of future climate scenarios under anthropogenic forcing is critical to understand climate change and to assess the impact of potentially counter-acting technologies. Machine learning and hybrid techniques for this prediction rely on informative metrics that are sensitive to pertinent but often subtle influences. For atmospheric dynamics, a critical part of the climate system, no well established metric exists and visual inspection is currently still often used in practice. However, this "eyeball metric" cannot be used for machine learning where an algorithmic description is required. Motivated by the success of intermediate neural network activations as basis for learned metrics, e.g. in computer vision, we present a novel, self-supervised representation learning approach specifically designed for atmospheric dynamics. Our approach, called AtmoDist, trains a neural network on a simple, auxiliary task: predicting the temporal distance between elements of a randomly shuffled sequence of atmospheric fields (e.g. the components of the wind field from reanalysis or simulation). The task forces the network to learn important intrinsic aspects of the data as activations in its layers and from these hence a discriminative metric can be obtained. We demonstrate this by using AtmoDist to define a metric for GAN-based super resolution of vorticity and divergence. Our upscaled data matches both visually and in terms of its statistics a high resolution reference closely and it significantly outperform the state-of-the-art based on mean squared error. Since AtmoDist is unsupervised, only requires a temporal sequence of fields, and uses a simple auxiliary task, it has the potential to be of utility in a wide range of applications.

cs.LG↗

arXiv2021

A neural network multigrid solver for the Navier-Stokes equations

We present the deep neural network multigrid solver (DNN-MG) that we develop for the instationary Navier-Stokes equations. DNN-MG improves computational efficiency using a judicious combination of a geometric multigrid solver and a recurrent neural network with memory. DNN-MG uses the multi-grid method to classically solve on coarse levels while the neural network corrects interpolated solutions on fine ones, thus avoiding the increasingly expensive computations that would have to be performed there. This results in a reduction in computation time through DNN-MG's highly compact neural network. The compactness results from its design for local patches and the available coarse multigrid solutions that provides a "guide" for the corrections. A compact neural network with a small number of parameters also reduces training time and data. Furthermore, the network's locality facilitates generalizability and allows one to use DNN-MG trained on one mesh domain also on different ones. We demonstrate the efficacy of DNN-MG for variations of the 2D laminar flow around an obstacle. For these, our method significantly improves the solutions as well as lift and drag functionals while requiring only about half the computation time of a full multigrid solution. We also show that DNN-MG trained for the configuration with one obstacle can be generalized to other time dependent problems that can be solved efficiently using a geometric multigrid method.

physics.comp-ph↗

arXiv2021

Deep neural networks for geometric multigrid methods

We investigate scaling and efficiency of the deep neural network multigrid method (DNN-MG). DNN-MG is a novel neural network-based technique for the simulation of the Navier-Stokes equations that combines an adaptive geometric multigrid solver, i.e. a highly efficient classical solution scheme, with a recurrent neural network with memory. The neural network replaces in DNN-MG one or multiple finest multigrid layers and provides a correction for the classical solve in the next time step. This leads to little degradation in the solution quality while substantially reducing the overall computational costs. At the same time, the use of the multigrid solver at the coarse scales allows for a compact network that is easy to train, generalizes well, and allows for the incorporation of physical constraints. Previous work on DNN-MG focused on the overall scheme and how to enforce divergence freedom in the solution. In this work, we investigate how the network size affects training and solution quality and the overall runtime of the computations. Our results demonstrate that larger networks are able to capture the flow behavior better while requiring only little additional training time. At runtime, the use of the neural network correction can even reduce the computation time compared to a classical multigrid simulation through a faster convergence of the nonlinear solve that is required at every time step.

math.NA↗

arXiv2020

Structure Preservation for the Deep Neural Network Multigrid Solver

The simulation of partial differential equations is a central subject of numerical analysis and an indispensable tool in science, engineering and related fields. Existing approaches, such as finite elements, provide (highly) efficient tools but deep neural network-based techniques emerged in the last few years as an alternative with very promising results. We investigate the combination of both approaches for the approximation of the Navier-Stokes equations and to what extent structural properties such as divergence freedom can and should be respected. Our work is based on DNN-MG, a deep neural network multigrid technique, that we introduced recently and which uses a neural network to represent fine grid fluctuations not resolved by a geometric multigrid finite element solver. Although DNN-MG provides solutions with very good accuracy and is computationally highly efficient, we noticed that the neural network-based corrections substantially violate the divergence freedom of the velocity vector field. In this contribution, we discuss these findings and analyze three approaches to address the problem: a penalty term to encourage divergence freedom of the network output; a penalty term for the corrected velocity field; and a network that learns the stream function, i.e. the scalar potential of the divergence free velocity vector field and which hence yields by construction divergence free corrections. Our experimental results show that the third approach based on the stream function outperforms the other two and not only improves the divergence freedom but in particular also the overall fidelity of the simulation.

math.NA↗

arXiv2020

A Local Spectral Exterior Calculus for the Sphere and Application to the Shallow Water Equations

We introduce $Ψ\mathrm{ec}$, a local spectral exterior calculus for the two-sphere $S^2$. $Ψ\mathrm{ec}$ provides a discretization of Cartan's exterior calculus on $S^2$ formed by spherical differential $r$-form wavelets. These are well localized in space and frequency and provide (Stevenson) frames for the homogeneous Sobolev spaces $\dot{H}^{-r+1}( Ω_ν^{r} , S^2 )$ of differential $r$-forms. At the same time, they satisfy important properties of the exterior calculus, such as the de Rahm complex and the Hodge-Helmholtz decomposition. Through this, $Ψ\mathrm{ec}$ is tailored towards structure preserving discretizations that can adapt to solutions with varying regularity. The construction of $Ψ\mathrm{ec}$ is based on a novel spherical wavelet frame for $L_2(S^2)$ that we obtain by introducing scalable reproducing kernel frames. These extend scalable frames to weighted sampling expansions and provide an alternative to quadrature rules for the discretization of needlet-like scale-discrete wavelets. We verify the practicality of $Ψ\mathrm{ec}$ for numerical computations using the rotating shallow water equations. Our numerical results demonstrate that a $Ψ\mathrm{ec}$-based discretization of the equations attains accuracy comparable to those of spectral methods while using a representation that is well localized in space and frequency.

math.NA↗

arXiv2019

Local Fourier Slice Photography

Light field cameras provide intriguing possibilities, such as post-capture refocus or the ability to synthesize images from novel viewpoints. This comes, however, at the price of significant storage requirements. Compression techniques can be used to reduce these but refocusing and reconstruction require so far again a dense pixel representation. To avoid this, we introduce local Fourier slice photography that allows for refocused image reconstruction directly from a sparse wavelet representation of a light field, either to obtain an image or a compressed representation of it. The result is made possible by wavelets that respect the "slicing's" intrinsic structure and enable us to derive exact reconstruction filters for the refocused image in closed form. Image reconstruction then amounts to applying these filters to the light field's wavelet coefficients, and hence no reconstruction of a dense pixel representation is required. We demonstrate that this substantially reduces storage requirements and also computation times. We furthermore analyze the computational complexity of our algorithm and show that it scales linearly with the size of the reconstructed region and the non-negligible wavelet coefficients, i.e. with the visual complexity.

cs.GR↗

arXiv2018

$\Psi$ec: A Local Spectral Exterior Calculus

We introduce $\Psi \mathrm{ec}$, a discretization of Cartan's exterior calculus of differential forms using wavelets. Our construction consists of differential $r$-form wavelets with flexible directional localization that provide tight frames for the spaces $\Omega^r(\mathbb{R}^n)$ of forms in $\mathbb{R}^2$ and $\mathbb{R}^3$. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the $k$-dimensional integral of an $r$-form is an $(r-k)$-form. They also verify Stokes' theorem for differential forms, with the most efficient finite dimensional approximation attained using directionally localized, curvelet- or ridgelet-like forms. The construction of $\Psi \mathrm{ec}$ builds on the geometric simplicity of the exterior calculus in the Fourier domain. We establish this structure by extending existing results on the Fourier transform of differential forms to a frequency description of the exterior calculus, including, for example, a Plancherel theorem for forms and a description of the symbols of all important operators.

math.NA↗

arXiv2018

A Local Fourier Slice Theorem

We present a local Fourier slice equation that enables local and sparse projection of a signal. Our result exploits that a slice in frequency space is an iso-parameter set in spherical coordinates. Therefore, the projection of suitable wavelets defined separably in these coordinates can be computed analytically, yielding a sequence of wavelets closed under projection. Our local Fourier slice equation then realizes projection as reconstruction with "sliced" wavelets with computational costs that scale linearly in the complexity of the projected signal. We numerically evaluate the performance of our local Fourier slice equation for synthetic test data and tomographic reconstruction, demonstrating that locality and sparsity can significantly reduce computation times and memory requirements.

math.NA↗

arXiv2018

Polar Wavelets in Space

Recent work introduced a unified framework for steerable and directional wavelets in two and three dimensions that ensures many desirable properties, such as a multi-scale structure, fast transforms, and a flexible angular localization. We show that, for an appropriate choice for the radial window function, these wavelets also have closed form expressions for, among other things, the spatial representation, the filter taps for the fast transform, and the frame representation of the Laplace operator. The numerical practicality and benefits of our work are demonstrated using signal estimation from non-uniform, point-wise samples, as required for example in ray tracing, and for reconstructing a signal over a lower-dimensional sub-manifold, with applications for instance in medical imaging.

math.NA↗

arXiv2018

Divergence Free Polar Wavelets for the Analysis and Representation of Fluid Flows

We present a Parseval tight wavelet frame for the representation and analysis of velocity vector fields of incompressible fluids. Our wavelets have closed form expressions in the frequency and spatial domains, are divergence free in the ideal, analytic sense, have a multi-resolution structure and fast transforms, and an intuitive correspondence to common flow phenomena. Our construction also allows for well defined directional selectivity, e.g. to model the behavior of divergence free vector fields in the vicinity of boundaries or to represent highly directional features like in a von K\'arm\'an vortex street. We demonstrate the practicality and efficiency of our construction by analyzing the representation of different divergence free vector fields in our wavelets.

math.NA↗

arXiv2017

Bendlets: A Second-Order Shearlet Transform with Bent Elements

We introduce bendlets, a shearlet-like system that is based on anisotropic scaling, translation, shearing, and bending of a compactly supported generator. With shearing being linear and bending quadratic in spatial coordinates, bendlets provide what we term a second-order shearlet system. As we show in this article, the decay rates of the associated transform enable the precise characterization of location, orientation and curvature of discontinuities in piecewise constant images. These results yield an improvement over existing directional representation systems where curvature only controls the constant of the decay rate of the transform. We also detail the construction of shearlet systems of arbitrary order. A practical implementation of bendlets is provided as an extension of the ShearLab toolbox, which we use to verify our theoretical classification results.

math.FA↗

arXiv2013

The Geometry of Radiative Transfer

We present the geometry and symmetries of radiative transfer theory. Our geometrization exploits recent work in the literature that enables to obtain the Hamiltonian formulation of radiative transfer as the semiclassical limit of a phase space representation of electromagnetic theory. Cosphere bundle reduction yields the traditional description over the space of "positions and directions", and geometrical optics arises as a special case when energy is disregarded. It is also shown that, in idealized environments, radiative transfer is a Lie-Poisson system with the group of canonical transformations as configuration space and symmetry group.

math-ph↗

arXiv2012

A Primer on Geometric Mechanics

Geometric mechanics is usually studied in applied mathematics and most introductory texts are hence aimed at a mathematically minded audience. The present note tries to provide the intuition of geometric mechanics and to show the relevance of the subject for an understanding of "mechanics".

math-ph↗