# Physics-informed deep learning and compressive collocation for high-dimensional diffusion-reaction equations: practical existence theory and numerics

arXiv: https://arxiv.org/abs/2406.01539 · HTML: https://mlsanity.com/paper/2406.01539/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 72 references that are in the map.
- First posted on arXiv 3 Jun 2024.

## Builds on (most cited first, 16 of 16)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning](https://mlsanity.com/paper/2008.13333/) (2020)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [The gap between theory and practice in function approximation with deep neural networks](https://mlsanity.com/paper/2001.07523/) (2020)
- [Deep Neural Networks Are Effective At Learning High-Dimensional Hilbert-Valued Functions From Limited Data](https://mlsanity.com/paper/2012.06081/) (2020)
- [A practical existence theorem for reduced order models based on convolutional autoencoders](https://mlsanity.com/paper/2402.00435/) (2024)
- [Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks](https://mlsanity.com/paper/2404.03761/) (2024)
- [Analysis of Deep Neural Networks with Quasi-optimal polynomial approximation rates](https://mlsanity.com/paper/1912.02302/) (2019)
- [Compressive Fourier collocation methods for high-dimensional diffusion equations with periodic boundary conditions](https://mlsanity.com/paper/2206.01255/) (2022)
- [Square Root LASSO: Well-posedness, Lipschitz stability and the tuning trade off](https://mlsanity.com/paper/2303.15588/) (2023)

## Built on by (most cited first, 2 of 2)

- [Beyond Universal Approximation Theorems: Algorithmic Uniform Approximation by Neural Networks Trained with Noisy Data](https://mlsanity.com/paper/2509.00924/) (2025)
- [Reconstruction of frequency-localized functions from pointwise samples via least squares and deep learning](https://mlsanity.com/paper/2502.09794/) (2025)

## Abstract (arXiv)

On the forefront of scientific computing, Deep Learning (DL), i.e., machine learning with Deep Neural Networks (DNNs), has emerged a powerful new tool for solving Partial Differential Equations (PDEs). It has been observed that DNNs are particularly well suited to weakening the effect of the curse of dimensionality, a term coined by Richard E. Bellman in the late `50s to describe challenges such as the exponential dependence of the sample complexity, i.e., the number of samples required to solve an approximation problem, on the dimension of the ambient space. However, although DNNs have been used to solve PDEs since the `90s, the literature underpinning their mathematical efficiency in terms of numerical analysis (i.e., stability, accuracy, and sample complexity), is only recently beginning to emerge. In this paper, we leverage recent advancements in function approximation using sparsity-based techniques and random sampling to develop and analyze an efficient high-dimensional PDE solver based on DL. We show, both theoretically and numerically, that it can compete with a novel stable and accurate compressive spectral collocation method for the solution of high-dimensional, steady-state diffusion-reaction equations with periodic boundary conditions. In particular, we demonstrate a new practical existence theorem, which establishes the existence of a class of trainable DNNs with suitable bounds on the network architecture and a sufficient condition on the sample complexity, with logarithmic or, at worst, linear scaling in dimension, such that the resulting networks stably and accurately approximate a diffusion-reaction PDE with high probability.