# A Theoretical Analysis of Deep Neural Networks and Parametric PDEs

arXiv: https://arxiv.org/abs/1904.00377 · HTML: https://mlsanity.com/paper/1904.00377/

## Key facts

- Cited by 74 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Mar 2019.

## Built on by (most cited first, 30 of 74)

- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
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- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations](https://mlsanity.com/paper/2112.08125/) (2021)
- [Operator learning with PCA-Net: upper and lower complexity bounds](https://mlsanity.com/paper/2303.16317/) (2023)
- [Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality](https://mlsanity.com/paper/2106.08900/) (2021)
- [Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems](https://mlsanity.com/paper/2212.03130/) (2022)
- [Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spaces](https://mlsanity.com/paper/2104.02746/) (2021)
- [wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws](https://mlsanity.com/paper/2207.08483/) (2022)
- [A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations](https://mlsanity.com/paper/2103.06183/) (2021)
- [Two-layer neural networks with values in a Banach space](https://mlsanity.com/paper/2105.02095/) (2021)
- [Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: comparison with linear subspace techniques](https://mlsanity.com/paper/2107.11460/) (2021)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Parametric Complexity Bounds for Approximating PDEs with Neural Networks](https://mlsanity.com/paper/2103.02138/) (2021)
- [Variational Physics Informed Neural Networks: the role of quadratures and test functions](https://mlsanity.com/paper/2109.02035/) (2021)
- [Learning Density Distribution of Reachable States for Autonomous Systems](https://mlsanity.com/paper/2109.06728/) (2021)
- [An artificial neural network approach to bifurcating phenomena in computational fluid dynamics](https://mlsanity.com/paper/2109.10765/) (2021)
- [CS4ML: A general framework for active learning with arbitrary data based on Christoffel functions](https://mlsanity.com/paper/2306.00945/) (2023)
- [Agnostic Active Learning of Single Index Models with Linear Sample Complexity](https://mlsanity.com/paper/2405.09312/) (2024)
- [Operator Learning of Lipschitz Operators: An Information-Theoretic Perspective](https://mlsanity.com/paper/2406.18794/) (2024)
- [Connections between Numerical Algorithms for PDEs and Neural Networks](https://mlsanity.com/paper/2107.14742/) (2021)
- [Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order models](https://mlsanity.com/paper/2201.10215/) (2022)
- [Multilevel CNNs for Parametric PDEs](https://mlsanity.com/paper/2304.00388/) (2023)
- [A practical existence theorem for reduced order models based on convolutional autoencoders](https://mlsanity.com/paper/2402.00435/) (2024)
- [Approximation results for Gradient Descent trained Shallow Neural Networks in 1d](https://mlsanity.com/paper/2209.08399/) (2022)
- [Generative Neural Operators of Log-Complexity Can Simultaneously Solve Infinitely Many Convex Programs](https://mlsanity.com/paper/2508.14995/) (2025)
- [Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimation](https://mlsanity.com/paper/2512.21319/) (2025)

## Abstract (arXiv)

We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent low-dimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical neural network approximation results. Concretely, we use the existence of a small reduced basis to construct, for a large variety of parametric partial differential equations, neural networks that yield approximations of the parametric solution maps in such a way that the sizes of these networks essentially only depend on the size of the reduced basis.