# Space-time deep neural network approximations for high-dimensional partial differential equations

arXiv: https://arxiv.org/abs/2006.02199 · HTML: https://mlsanity.com/paper/2006.02199/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jun 2020.

## Built on by (most cited first, 9 of 9)

- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems](https://mlsanity.com/paper/2212.03130/) (2022)
- [Spline parameterization of neural network controls for deep learning](https://mlsanity.com/paper/2103.00301/) (2021)
- [Approximation properties of Residual Neural Networks for Kolmogorov PDEs](https://mlsanity.com/paper/2111.00215/) (2021)
- [Lagrangian Density Space-Time Deep Neural Network Topology](https://mlsanity.com/paper/2207.12209/) (2022)
- [Transferability of Winning Lottery Tickets in Neural Network Differential Equation Solvers](https://mlsanity.com/paper/2306.09863/) (2023)
- [Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximation](https://mlsanity.com/paper/2607.27781/) (2026)
- [Residual neural networks overcome the curse of dimensionality for semilinear heat equations](https://mlsanity.com/paper/2609.03626/) (2026)

## Abstract (arXiv)

It is one of the most challenging issues in applied mathematics to approximately solve high-dimensional partial differential equations (PDEs) and most of the numerical approximation methods for PDEs in the scientific literature suffer from the so-called curse of dimensionality in the sense that the number of computational operations employed in the corresponding approximation scheme to obtain an approximation precision varepsilon>0 grows exponentially in the PDE dimension and/or the reciprocal of varepsilon. Recently, certain deep learning based approximation methods for PDEs have been proposed and various numerical simulations for such methods suggest that deep neural network (DNN) approximations might have the capacity to indeed overcome the curse of dimensionality in the sense that the number of real parameters used to describe the approximating DNNs grows at most polynomially in both the PDE dimension d∈mathbbN and the reciprocal of the prescribed accuracy varepsilon>0. There are now also a few rigorous results in the scientific literature which substantiate this conjecture by proving that DNNs overcome the curse of dimensionality in approximating solutions of PDEs. Each of these results establishes that DNNs overcome the curse of dimensionality in approximating suitable PDE solutions at a fixed time point T>0 and on a compact cube [a,b]d in space but none of these results provides an answer to the question whether the entire PDE solution on [0,T]× [a,b]d can be approximated by DNNs without the curse of dimensionality. It is precisely the subject of this article to overcome this issue. More specifically, the main result of this work in particular proves for every a∈ℝ,  b∈ (a,∞) that solutions of certain Kolmogorov PDEs can be approximated by DNNs on the space-time region [0,T]× [a,b]d without the curse of dimensionality.