# Data-driven discovery of Green's functions

arXiv: https://arxiv.org/abs/2210.16016 · HTML: https://mlsanity.com/paper/2210.16016/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 247 references that are in the map.
- First posted on arXiv 28 Oct 2022.

## Builds on (most cited first, 10 of 10)

- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Searching for Activation Functions](https://mlsanity.com/paper/1710.05941/) (2017)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Polynomial Regression As an Alternative to Neural Nets](https://mlsanity.com/paper/1806.06850/) (2018)
- [A Polynomial-Based Approach for Architectural Design and Learning with Deep Neural Networks](https://mlsanity.com/paper/1905.10457/) (2019)

## Abstract (arXiv)

Discovering hidden partial differential equations (PDEs) and operators from data is an important topic at the frontier between machine learning and numerical analysis. This doctoral thesis introduces theoretical results and deep learning algorithms to learn Green's functions associated with linear partial differential equations and rigorously justify PDE learning techniques. A theoretically rigorous algorithm is derived to obtain a learning rate, which characterizes the amount of training data needed to approximately learn Green's functions associated with elliptic PDEs. The construction connects the fields of PDE learning and numerical linear algebra by extending the randomized singular value decomposition to non-standard Gaussian vectors and Hilbert--Schmidt operators, and exploiting the low-rank hierarchical structure of Green's functions using hierarchical matrices. Rational neural networks (NNs) are introduced and consist of neural networks with trainable rational activation functions. The highly compositional structure of these networks, combined with rational approximation theory, implies that rational functions have higher approximation power than standard activation functions. In addition, rational NNs may have poles and take arbitrarily large values, which is ideal for approximating functions with singularities such as Green's functions. Finally, theoretical results on Green's functions and rational NNs are combined to design a human-understandable deep learning method for discovering Green's functions from data. This approach complements state-of-the-art PDE learning techniques, as a wide range of physics can be captured from the learned Green's functions such as dominant modes, symmetries, and singularity locations.