We propose a deep learning based method, the Deep Ritz Method, for numerically solving variational problems, particularly the ones that arise from partial differential equations. The Deep Ritz method is naturally nonlinear, naturally adaptive and has the potential to work in rather high dimensions. The framework is qui…
Paper uses deep Ritz method for solving stationary Schrödinger equation, proving convergence and feature emergence.
problem Solving stationary Schrödinger equation with high-dimensional features.
method Deep Ritz method, gradient descent, single-index model, two-neuron model.
result Gradient descent converges to near-optimal solution, feature emergence observed in two-neuron model.
Paper studies deep learning for solving elliptic PDEs, proving optimal bounds and neural scaling laws.
problem Solving elliptic PDEs from random samples using machine learning.
method Deep Ritz Method and Physics-Informed Neural Networks (PINNs) for the Schrödinger equation.
result Proves minimax optimal bounds and neural scaling laws for deep PDE solvers.
Paper analyzes DRM for solving high-dimensional elliptic PDEs with generalization bounds.
problem Analyzing generalization error of neural network methods for high-dimensional PDEs.
method Developed a new solution theory for spectral Barron space and derived generalization error bounds.
result Generalization error bounds are independent of dimension and solutions lie in spectral Barron space.
Study analyzes error in neural network solving PDEs, providing convergence and error bounds.
problem Error analysis of neural network solving PDEs.
method Three-layer tanh neural network with projected gradient descent (PGD).
result Comprehensive error analysis including approximation, generalization, and optimization errors.
This paper compares deeper and wider neural networks for optimal generalization error in Sobolev losses.
problem The dilemma of choosing between deeper or wider neural networks for optimal generalization error.
method Analytical investigations into the influence of sample points, parameters, and loss function regularity on neural network architecture.
result A higher number of parameters favors wider neural networks, while more sample points and greater loss function regularity favor deeper neural networks.
Paper develops IFTRR to solve sparse generalized eigenvalue problems efficiently.
problem Finding the leading eigenvector with at most k nonzero entries in sparse generalized eigenvalue problems.
method Inverse-free truncated Rayleigh-Ritz method (IFTRR) with a new truncation strategy.
result IFTRR efficiently finds the support set of the leading eigenvector for large scale problems.
In this paper, we propose the idea of radial scaling in frequency domain and activation functions with compact support to produce a multi-scale DNN (MscaleDNN), which will have the multi-scale capability in approximating high frequency and high dimensional functions and speeding up the solution of high dimensional PDEs…
New neural network approach solves Poisson equations efficiently.
problem Approximating solutions to Poisson equations with Dirichlet boundary conditions.
method Using shallow ReLUα-networks to solve Laplace operator equations. result Neural networks can approximate solutions to the Laplace operator with Dirichlet boundary conditions efficiently.
For a bounded domain Ω with a piecewise smooth boundary in a complete Riemannian manifold M, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of L2(Ω) in place of the Rayleigh-Ritz formula, we obtain inequalities for …
NWoS solves high-dimensional Poisson equations using neural networks.
problem Efficiently solving high-dimensional Poisson equations.
method Neural Walk-on-Spheres (NWoS) leveraging stochastic representations and Walk-on-Spheres methods.
result NWoS outperforms competing methods in accuracy, speed, and computational costs.
We produce a new proof and extend results by Harrell and Stubbe for the discrete spectrum of a self-adjoint operator. An abstract approach--based on commutator algebra, the Rayleigh-Ritz principle, and an ``optimal'' usage of the Cauchy-Schwarz inequality--is used to produce ``parameter-free'', ``projection-free'' vers…
Gradient descent achieves optimal learning for elliptic PDEs via Sobolev norms.
problem Learning elliptic PDEs from noisy data.
method Gradient descent on Sobolev norm objective functions.
result Gradient descent achieves statistical optimality for elliptic PDEs.
Fast and accurate methods for low-rank learning problems.
problem Partial singular value decomposition and numerical rank estimation of huge matrices.
method Krylov subspaces and Ritz vectors for fast and accurate solutions.
result Advantages over traditional methods in accuracy and speed.
Stochastic volatility (SV) and local stochastic volatility (LSV) processes can be used to model the evolution of various financial variables such as FX rates, stock prices, and so on. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes. Many issues remain, thou…
New algorithm updates eigenvectors of evolving graphs efficiently.
problem Updating eigenvectors of dynamic graphs.
method Subspace projection based on Rayleigh-Ritz projections.
result Strong performance in eigenvector approximation and downstream tasks.
Probabilistic deep learning uses neural networks and models to handle uncertainty.
problem Handling uncertainty in deep learning models.
method Two approaches: probabilistic neural networks and deep probabilistic models.
result TensorFlow Probability library supports both approaches.
Improves deep learning robustness by considering task and model.
problem Adversarial attacks on deep learning systems.
method Binary and interval label encoding strategy to redefine classification tasks and design corresponding loss functions.
result Our method enhances robustness without sacrificing accuracy.
Deep-RLS uses deep learning to improve PCA for better source separation.
problem Improving PCA for better source separation in nonlinear systems.
method Inspired by RLS, Deep-RLS unfolds RLS iterations into a deep neural network.
result Deep-RLS significantly improves accuracy in recovering source signals.
How to understand deep learning systems remains an open problem. In this paper we propose that the answer may lie in the geometrization of deep networks. Geometrization is a bridge to connect physics, geometry, deep network and quantum computation and this may result in a new scheme to reveal the rule of the physical w…
Deep learning methods improve overlapping speaker separation across languages and noise.
problem Overlapping speaker separation in realistic scenarios.
method Deep clustering and deep attractor networks.
result Deep learning methods are effective for a broad range of languages and can handle untrained languages with common features.
SDF adapts Deep Forest for evolving data streams with active learning.
problem Adapting Deep Forest for evolving data streams.
method Streaming Deep Forest (SDF) with Augmented Variable Uncertainty (AVU) active learning.
result SDF with AVU outperforms other methods trained with all instances by 70% labeling budget.
Deep SSMs use neural networks to identify complex systems.
problem Identifying nonlinear systems with high uncertainty.
method Deep state space models with neural networks.
result Deep SSMs outperform traditional methods on benchmarks.
This paper presents a basic property of region dividing of ReLU (rectified linear unit) deep learning when new layers are successively added, by which two new perspectives of interpreting deep learning are given. The first is related to decision trees and forests; we construct a deep learning structure equivalent to a …
The great success of deep learning shows that its technology contains profound truth, and understanding its internal mechanism not only has important implications for the development of its technology and effective application in various fields, but also provides meaningful insights into the understanding of human brai…
DSCF-Net learns deep features for clustering with robustness and locality preservation.
problem Unsupervised deep representation learning for clustering.
method Integrates robust deep concept factorization, deep self-expressive representation, and adaptive locality preserving feature learning.
result Delivers state-of-the-art performance on public databases.
Bayesian methods enhance deep learning models by improving reliability and uncertainty.
problem Improving reliability and uncertainty awareness in deep learning models.
method Approximate Bayesian inference techniques, including SG-MCMC and VI, applied to deep learning models.
result Enhanced posterior inference for deep learning models, particularly in neural networks and generative models.
NeurIPS 2020 competition seeks to predict deep learning generalization.
problem Understanding and predicting generalization in deep learning models.
method Propose complexity measures to accurately predict generalization performance.
result A robust complexity measure could improve deep learning reliability.
This paper analyzes generalization issues in deep reinforcement learning.
problem Understanding and improving generalization capabilities of deep reinforcement learning policies.
method Formalizing and categorizing solutions to address overfitting in deep reinforcement learning.
result A comprehensive analysis of generalization challenges and solutions in deep reinforcement learning.
This paper explains why ResNets generalize better than FFNets using neural tangent kernels.
problem Understanding why deep ResNets generalize better than deep FFNets.
method Using neural tangent kernels to compare the learnability of functions induced by the kernels of ResNets and FFNets.
result The kernel of ResNets does not exhibit degeneracy as depth increases, unlike FFNets.
This paper provides an overview of deep semi-supervised learning methods.
problem Reducing the need for large annotated datasets in deep learning.
method Summarizes dominant semi-supervised approaches in deep learning.
result Provides a comprehensive overview of deep semi-supervised learning.
In this paper, we have proposed a deep quantum SVM formulation, and further demonstrated a quantum-clustering framework based on the quantum deep SVM formulation, deep convolutional neural networks, and quantum K-Means clustering. We have investigated the run time computational complexity of the proposed quantum deep c…
Deep learning is very effective at jointly learning feature representations and classification models, especially when dealing with high dimensional input patterns. Probabilistic logic reasoning, on the other hand, is capable to take consistent and robust decisions in complex environments. The integration of deep learn…
With the growth of deep learning, how to describe deep neural networks unifiedly is becoming an important issue. We first formalize neural networks mathematically with their directed graph representations, and prove a generation theorem about the induced networks of connected directed acyclic graphs. Then, we set up a …
Deep active inference learns policies from sensory inputs.
problem Learning policies in partially observable domains.
method Optimizes expected free energy with a variational autoencoder.
result Comparable or better performance than deep Q-learning.
Deep learning aids causal inference in complex settings.
problem Estimating heterogeneous treatment effects in non-linear, time-varying, and encoded confounders.
method Intuitive introduction to deep learning and causal inference, focusing on observational data.
result Maximizes accessibility to causal inference through deep learning.
Deep learning is increasingly being used in high-stake decision making applications that affect individual lives. However, deep learning models might exhibit algorithmic discrimination behaviors with respect to protected groups, potentially posing negative impacts on individuals and society. Therefore, fairness in deep…
This paper introduces Deep Incremental Boosting, a new technique derived from AdaBoost, specifically adapted to work with Deep Learning methods, that reduces the required training time and improves generalisation. We draw inspiration from Transfer of Learning approaches to reduce the start-up time to training each incr…
Proposes model-based robust deep learning to handle natural variation in data.
problem Deep learning's fragility to natural variation in data.
method Develops model-based robust training algorithms using deep generative models to learn natural variation.
result Deep neural networks trained with model-based algorithms outperform standard and norm-bounded robust algorithms.
Deep learning networks are approximated using dynamical systems theory.
problem Understanding the approximation capabilities of deep learning networks.
method Modeling deep residual networks as continuous-time dynamical systems and using approximation theories in Lp. result Established general sufficient conditions for universal approximation of deep residual networks.
In this paper we introduce ZhuSuan, a python probabilistic programming library for Bayesian deep learning, which conjoins the complimentary advantages of Bayesian methods and deep learning. ZhuSuan is built upon Tensorflow. Unlike existing deep learning libraries, which are mainly designed for deterministic neural netw…
Deep ReLU networks can approximate matrix-vector products with error bounds.
problem Can deep ReLU networks accurately approximate matrix-vector products?
method Derived error bounds in Lebesgue and Sobolev norms for deep ReLU FNNs.
result Developed deep approximation theory with successful applications.
Deep RL applied for Indian stock trading strategies.
problem Designing profitable trading strategies for Indian stock markets.
method Applied deep reinforcement learning to ten Indian stock datasets.
result Models' performance compared and evaluated.
Hashing has been widely used for large-scale approximate nearest neighbor search because of its storage and search efficiency. Recent work has found that deep supervised hashing can significantly outperform non-deep supervised hashing in many applications. However, most existing deep supervised hashing methods adopt a …
Deep learning methods are reviewed for preserving structure in neural networks.
problem Challenges in applying deep learning, especially in preserving structure.
method Review of existing deep learning methods and new algorithmic frameworks.
result Mathematical understanding and systematic design of deep learning methods to preserve structure.
Characterizes deep neural network weight space for adversarial attacks.
problem Poor performance of deep learning models in adversarial examples.
method Characterizes deep neural network solution space using two paradigms.
result Adversarial attacks are less successful against Associative Memory Models.
Although deep learning has shown great success in recent years, researchers have discovered a critical flaw where small, imperceptible changes in the input to the system can drastically change the output classification. These attacks are exploitable in nearly all of the existing deep learning classification frameworks.…
Bayesian interpretation of deep ensembles improves uncertainty quantification.
problem Improving uncertainty estimation in deep learning models.
method Viewing deep ensembles as an approximate Bayesian method and specifying corresponding assumptions.
result Improved approximation leads to larger epistemic uncertainty, potentially more reliable predictions.