Search Results for author:
Found 32 papers, 8 papers with code
Sampling Complexity of Deep Approximation Spaces
While it is well-known that neural networks enjoy excellent approximation capabilities, it remains a big challenge to compute such approximations from point samples.
Variational Monte Carlo on a Budget -- Fine-tuning pre-trained Neural Wavefunctions
Obtaining accurate solutions to the Schr\"odinger equation is the key challenge in computational quantum chemistry.
FakET: Simulating Cryo-Electron Tomograms with Neural Style Transfer
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.
Towards a Foundation Model for Neural Network Wavefunctions
Furthermore, we provide ample experimental evidence to support the idea that extensive pre-training of a such a generalized wavefunction model across different compounds and geometries could lead to a foundation wavefunction model.
Learning ReLU networks to high uniform accuracy is intractable
Statistical learning theory provides bounds on the necessary number of training samples needed to reach a prescribed accuracy in a learning problem formulated over a given target class.
Gold-standard solutions to the Schrödinger equation using deep learning: How much physics do we need?
Finding accurate solutions to the Schr\"odinger equation is the key unsolved challenge of computational chemistry.
Integral representations of shallow neural network with Rectified Power Unit activation function
In this effort, we derive a formula for the integral representation of a shallow neural network with the Rectified Power Unit activation function.
Sobolev-type embeddings for neural network approximation spaces
We consider neural network approximation spaces that classify functions according to the rate at which they can be approximated (with error measured in $L^p$) by ReLU neural networks with an increasing number of coefficients, subject to bounds on the magnitude of the coefficients and the number of hidden layers.
Solving the electronic Schrödinger equation for multiple nuclear geometries with weight-sharing deep neural networks
Accurate numerical solutions for the Schr\"odinger equation are of utmost importance in quantum chemistry.
The Modern Mathematics of Deep Learning
We describe the new field of mathematical analysis of deep learning.
Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spaces
Such algorithms (most prominently stochastic gradient descent and its variants) are used extensively in the field of deep learning.
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.
Approximations with deep neural networks in Sobolev time-space
Solutions of evolution equation generally lies in certain Bochner-Sobolev spaces, in which the solution may has regularity and integrability properties for the time variable that can be different for the space variables.
Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning
We show that a single deep neural network trained on simulated data is capable of learning the solution functions of an entire family of PDEs on a full space-time region.
Phase Transitions in Rate Distortion Theory and Deep Learning
We also provide quantitative and non-asymptotic bounds on the probability that a random $f\in\mathcal{S}$ can be encoded to within accuracy $\varepsilon$ using $R$ bits.
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.
Uniform error estimates for artificial neural network approximations for heat equations
These mathematical results from the scientific literature prove in part that algorithms based on ANNs are capable of overcoming the curse of dimensionality in the numerical approximation of high-dimensional PDEs.
Deep neural network approximations for Monte Carlo algorithms
One key argument in most of these results is, first, to use a Monte Carlo approximation scheme which can approximate the solution of the PDE under consideration at a fixed space-time point without the curse of dimensionality and, thereafter, to prove that DNNs are flexible enough to mimic the behaviour of the used approximation scheme.
Space-time error estimates for deep neural network approximations for differential equations
It is the subject of the main result of this article to provide space-time error estimates for DNN approximations of Euler approximations of certain perturbed differential equations.
How degenerate is the parametrization of neural networks with the ReLU activation function?
Approximation capabilities of neural networks can be used to deal with the latter non-convexity, which allows us to establish that for sufficiently large networks local minima of a regularized optimization problem on the realization space are almost optimal.
Towards a regularity theory for ReLU networks -- chain rule and global error estimates
Although for neural networks with locally Lipschitz continuous activation functions the classical derivative exists almost everywhere, the standard chain rule is in general not applicable.
The Oracle of DLphi
We present a novel technique based on deep learning and set theory which yields exceptional classification and prediction results.
Deep Neural Network Approximation Theory
This paper develops fundamental limits of deep neural network learning by characterizing what is possible if no constraints are imposed on the learning algorithm and on the amount of training data.
Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations
It can be concluded that ERM over deep neural network hypothesis classes overcomes the curse of dimensionality for the numerical solution of linear Kolmogorov equations with affine coefficients.
A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
Such numerical simulations suggest that ANNs have the capacity to very efficiently approximate high-dimensional functions and, especially, indicate that ANNs seem to admit the fundamental power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named computational problems.
The universal approximation power of finite-width deep ReLU networks
We show that finite-width deep ReLU neural networks yield rate-distortion optimal approximation (B\"olcskei et al., 2018) of polynomials, windowed sinusoidal functions, one-dimensional oscillatory textures, and the Weierstrass function, a fractal function which is continuous but nowhere differentiable.
Solving the Kolmogorov PDE by means of deep learning
Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences.
Topology Reduction in Deep Convolutional Feature Extraction Networks
Finally, for networks based on Weyl-Heisenberg filters, we determine the prototype function bandwidth that minimizes---for fixed network depth $N$---the average number of operationally significant nodes per layer.
Optimal Approximation with Sparsely Connected Deep Neural Networks
Specifically, all function classes that are optimally approximated by a general class of representation systems---so-called \emph{affine systems}---can be approximated by deep neural networks with minimal connectivity and memory requirements.
Energy Propagation in Deep Convolutional Neural Networks
This paper establishes conditions for energy conservation (and thus for a trivial null-set) for a wide class of deep convolutional neural network-based feature extractors and characterizes corresponding feature map energy decay rates.
Discrete Deep Feature Extraction: A Theory and New Architectures
First steps towards a mathematical theory of deep convolutional neural networks for feature extraction were made---for the continuous-time case---in Mallat, 2012, and Wiatowski and B\"olcskei, 2015.
Deep Convolutional Neural Networks on Cartoon Functions
Wiatowski and B\"olcskei, 2015, proved that deformation stability and vertical translation invariance of deep convolutional neural network-based feature extractors are guaranteed by the network structure per se rather than the specific convolution kernels and non-linearities.
