Neural networks can approximate functions uniformly across various measures.
problem Universal approximation of functions across different probability measures.
method Proving neural networks are dense in Orlicz spaces, extending classical theorems.
result Neural networks uniformly approximate functions for weakly compact families of measures.
Neural networks can approximate complex stochastic equations well.
problem Approximating general stochastic differential equations.
method Identified neural network classes approximating continuous functions.
result Neural stochastic differential equations can approximate general stochastic differential equations arbitrarily well.
Dense neural networks can't approximate all functions.
problem Approximation capabilities of dense neural networks.
method Model compression approach combining weak regularity lemma and graph neural networks.
result Existence of Lipschitz continuous functions not approximable by dense neural networks.
Deep neural networks with piecewise-polynomial activations can approximate smooth functions and their derivatives.
problem Approximating smooth functions and their derivatives with neural networks.
method Derives the depth, width, and sparsity required for approximation in Hölder norms.
result Deep neural networks with bounded weights can approximate Hölder smooth functions and their derivatives.
Review of neural network expressivity and architectures.
problem Understanding neural network expressivity across different architectures.
method Comprehensive overview of approximation results for various neural network types.
result Deep neural networks offer advantages over shallow ones for specific function classes.
Neural network based approximate computing is a universal architecture promising to gain tremendous energy-efficiency for many error resilient applications. To guarantee the approximation quality, existing works deploy two neural networks (NNs), e.g., an approximator and a predictor. The approximator provides the appro…
Paper proves neural networks can be approximated using interval bounds.
problem Verifying safety and robustness of neural networks.
method Introduces interval universal approximation (IUA) theorem for neural networks.
result Neural networks can be approximated using interval bounds for any continuous function and squashable activation functions.
Deep neural networks approximate functions in shift-invariant spaces with controlled error.
problem Approximating functions in shift-invariant spaces with neural networks.
method Using deep ReLU neural networks, estimating approximation error bounds based on network width and depth.
result Deep neural networks achieve optimal approximation rates for Sobolev spaces up to a logarithmic factor.
Formula found for neural network error with fixed weights.
problem Understanding error in neural networks with fixed weights.
method Provided an explicit formula for approximation error.
result Explicit formula for neural network error with fixed weights.
Gradient descent trains shallow neural networks to approximate functions in 1D.
problem Approximating functions in 1D with shallow neural networks trained by gradient descent.
method Gradient descent optimization of non-convex weight space for finite width networks in 1D.
result Gradient descent can approximate functions in 1D with a minimal number of weights, balancing practical performance and theoretical capabilities.
This work improves sample efficiency in neural function approximation for reinforcement learning.
problem Improving sample efficiency in reinforcement learning with neural function approximation.
method Study of function approximation with two-layer neural networks (ReLU and polynomial activations) under generative and realizability models.
result Significant improvement in sample complexity compared to linear methods.
Study shows limits on deep and shallow neural networks for approximating compact sets.
problem Understanding the limitations of deep and shallow neural networks in approximating compact sets.
method Proved Carl's type inequalities for approximation error, using Lipschitz widths.
result Lower bounds on approximation error for neural network outputs.
The paper bounds neural networks' approximation error and applies it to regression and GANs.
problem Bounding the approximation error of norm-constrained neural networks.
method Proved upper and lower bounds on approximation error using Rademacher complexity.
result Obtained convergence rates for over-parameterized neural networks and optimal GAN learning rates.
Quantum neural networks can approximate noisy functions accurately.
problem Approximating noisy functions with quantum neural networks.
method Universal approximation theorem with error bounds for noisy quantum neural networks.
result Quantum neural networks can approximate noisy functions with precise error bounds.
The paper shows neural networks can approximate functions over non-compact domains with non-polynomial activation.
problem Approximating functions over non-compact domains using neural networks.
method Using single-hidden-layer feedforward neural networks with non-polynomial activation functions over non-compact subsets of Euclidean spaces.
result Neural networks can approximate functions in weighted Ck-spaces and weighted Sobolev spaces over unbounded domains. Neural networks approximate superhedging prices in financial models.
problem Approximating superhedging prices in financial markets.
method Neural networks for approximating α-quantile hedging prices and their essential supremum. result Neural networks provide an approximation for superhedging prices and strategies.
Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.
problem Quantifying the efficiency of neural operators for solving nonlinear parabolic PDEs.
method Deriving approximation rates by transferring PDEs to integral equations and leveraging Picard's iteration.
result Neural operators can efficiently approximate solution operators of nonlinear PDEs without exponential complexity growth.
Paper shows deep neural networks can approximate Korobov functions nearly optimally.
problem Approximating Korobov functions with deep neural networks.
method Used deep neural networks and measured approximation rates with Lp and H1 norms. result Achieved a super-convergence rate, outperforming traditional methods.
Deep neural networks and the ENO procedure are both efficient frameworks for approximating rough functions. We prove that at any order, the ENO interpolation procedure can be cast as a deep ReLU neural network. This surprising fact enables the transfer of several desirable properties of the ENO procedure to deep neural…
TVS-FNNs can approximate any continuous function on expanded input spaces.
problem Processing a broader range of inputs like sequences and matrices.
method Proving a universal approximation theorem for TVS-FNNs.
result TVS-FNNs can approximate any continuous function on expanded input spaces.
We consider in this paper the optimal approximations of convex univariate functions with feed-forward Relu neural networks. We are interested in the following question: what is the minimal approximation error given the number of approximating linear pieces? We establish the necessary and sufficient conditions and uniqu…
Survey examines deep neural networks' ability to approximate functions.
problem Approximation of target functions by deep neural networks.
method Examination of feed-forward and residual architectures, focusing on optimization problems in regression and classification.
result Deep neural networks can approximate functions effectively, especially with ReLU activation functions.
Deep neural nets approximate high-dimensional HJB equations efficiently.
problem Approximating solutions to high-dimensional HJB equations.
method Deep neural networks for approximating solutions.
result Deep neural networks can approximate solutions without the curse of dimensionality.
Study assesses neural nets for optimization problems, highlighting SiLU's effectiveness.
problem Using neural nets for optimization problems, especially for accurate approximations.
method Determined best activation function (SiLU) for nonlinear optimization problems. Analyzed function approximations using neural networks and interpolation/regression models.
result Neural nets can deliver competitive zero- and first-order approximations but underperform on second-order approximations.
Quantum neural networks approximate periodic functions more efficiently.
problem Approximating periodic functions with quantum neural networks.
method Using Jackson's inequality to construct a QNN that approximates a trigonometric polynomial of the function.
result Quantum neural networks can achieve better approximation results with fewer parameters for smoother functions.
The study of universal approximation of arbitrary functions f:X→Y by neural networks has a rich and thorough history dating back to Kolmogorov (1957). In the case of learning finite dimensional maps, many authors have shown various forms of the universality of both fixed depth and fixed width…
The paper provides approximation guarantees for neural networks trained with gradient flow.
problem Approximating neural networks trained with gradient flow in continuous L2(Sd−1)-norm. method NTK argument for non-convex second but last layer, under-parametrized regime.
result Gradient flow convergence guarantees for neural networks under Sobolev smoothness assumptions.
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.
Q-learning with neural network function approximation (neural Q-learning for short) is among the most prevalent deep reinforcement learning algorithms. Despite its empirical success, the non-asymptotic convergence rate of neural Q-learning remains virtually unknown. In this paper, we present a finite-time analysis of a…
Optimal rates for shallow ReLU networks in nonparametric regression.
problem Approximating smooth and non-smooth functions with shallow ReLU networks.
method Analysis of shallow ReLUk neural networks, using variation norms and deep learning theory. result Optimal approximation rates for shallow ReLU networks in nonparametric regression.
The paper analyzes deep ReLU CNNs' approximation properties in 2D space.
problem Establishing L2 approximation properties for deep ReLU CNNs. method Analysis based on decomposition theorem for convolutional kernels, properties of ReLU activation, and connections with one-hidden-layer ReLU NNs.
result Universal approximation theorem for deep ReLU CNNs with classic structure.
This study shows neural nets can approximate Turing machines with meaningful statistical properties.
problem Theoretical limitations in approximating Turing machines with neural networks.
method Formal definition of statistically meaningful approximation, analysis of boolean circuits and Turing machines using neural nets.
result Transformers can statistically meaningfully approximate Turing machines with polynomial sample complexity.
Improved neural network approximates analytic and L^p functions efficiently.
problem Efficiently approximating analytic and L^p functions using neural networks.
method Three-dimensional ReLU network architecture for sawtooth functions, improving approximation rates.
result Substantially improved exponential approximation rates for analytic functions and general L^p functions.
This study explains why approximate NGD works well in wide neural networks.
problem Understanding why NGD with approximate Fisher information converges fast in wide neural networks.
method Analyzing asymptotic training dynamics in function space via the neural tangent kernel.
result NGD with approximate Fisher information achieves the same fast convergence as exact NGD under specific conditions.
Poor approximators found in neural networks and random feature models.
problem Understanding why certain neural networks and models perform poorly in approximating functions.
method Established a scale separation of Kolmogorov width type and applied it to neural networks and random feature models.
result Reproducing kernel Hilbert spaces and two-layer neural networks are poor L2-approximators for certain functions. Study shows how to approximate and estimate high-dimensional classification functions without the curse of dimensionality.
problem Approximating and estimating classification functions in high-dimensional spaces.
method Modified existing results to show that RBV2 functions can be approximated by neural networks with bounded weights. Proved the existence of a neural network with bounded weights approximating a classification function. Leveraged these bounds to quantify estimation rates. result Neural networks can approximate RBV2 functions without the curse of dimensionality, leading to efficient estimation rates. Deep neural nets approximate random dynamical system trajectories uniformly in time.
problem Approximating trajectories of random dynamical systems over infinite time horizons.
method Recurrent neural networks with simple feedback structures.
result Certain random trajectories can be approximated uniformly in time to any desired accuracy.
Neural operators solve families of 2BSDEs efficiently.
problem Solving infinite families of 2BSDEs on bounded domains.
method Introduces a mild generative neural operator model to approximate solutions.
result Solution operators can be approximated by neural operators with polynomial parameters.
Develops wavelet-based neural network approximation theory.
problem Analyzing neural network approximation capabilities over various activation functions.
method Wavelet frame theory on spaces of homogeneous type, sufficient conditions for approximation, error estimates.
result Derives sufficient conditions for neural networks to approximate any functions in a given space, including non-smooth activations.
Neural approximate computing gains enormous energy-efficiency at the cost of tolerable quality-loss. A neural approximator can map the input data to output while a classifier determines whether the input data are safe to approximate with quality guarantee. However, existing works cannot maximize the invocation of the a…
The paper proves deep neural networks with analytic activation can approximate any function.
problem Approximating functions with neural networks using analytic activation functions.
method Elementary proofs for real and complex networks, Stone-Weierstrass theorem, Mergelyan's theorem.
result Closure of neural network classes equals space of polynomials for analytic activation.
Single-spike neurons can approximate as well as multi-spike neurons.
problem Limitation of single-spike neurons in spiking neural networks.
method Comparison of single-spike and multi-spike neural networks.
result Single-spike and multi-spike neural networks are equivalent in approximation capabilities.
There has been a growing interest in expressivity of deep neural networks. However, most of the existing work about this topic focuses only on the specific activation function such as ReLU or sigmoid. In this paper, we investigate the approximation ability of deep neural networks with a broad class of activation functi…
Neural networks with learned biases can approximate any function.
problem Whether neural networks with only learned biases can approximate any continuous function.
method Theoretical and numerical analysis of random weights and learned biases in neural networks.
result Feedforward and recurrent neural networks with random weights can approximate any continuous function and dynamical systems.
Efficiently approximates neural network function space distance.
problem Estimating the average discrepancy between neural network outputs.
method Linearized Activation Function TRick (LAFTR) for ReLU networks.
result Parametric approximation outperforms nonparametric methods in memory and accuracy.
Researchers develop neural networks for approximating functions in Banach spaces.
problem Approximating Banach space valued continuous functions.
method Quasi-interpolation Banach space valued neural network operators using algebraic sigmoid functions.
result Jackson type inequalities for function approximation.
Deep networks can learn functions approximated by shallow networks, but not all functions.
problem The learnability of functions by deep neural networks and the approximation capacity of simpler classes.
method Study the connection between learnability and approximation capacity of functions by deep neural networks and simpler classes.
result A necessary condition for a function to be learnable by deep neural networks is to be approximable by shallow networks.
The paper shows how neural networks can approximate PDEs with polynomial scaling in dimension.
problem Understanding the complexity of approximating PDE solutions with neural networks.
method Developed a proof technique to simulate gradient descent using neural networks.
result Neural network parameters scale polynomially with input dimension for approximating PDE solutions.