# Bayesian Learning to Discover Mathematical Operations in Governing Equations of Dynamic Systems

arXiv: https://arxiv.org/abs/2206.00669 · HTML: https://mlsanity.com/paper/2206.00669/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 56 references that are in the map.
- First posted on arXiv 1 Jun 2022.

## Builds on (most cited first, 14 of 14)

- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Deep Compression: Compressing Deep Neural Networks with Pruning, Trained Quantization and Huffman Coding](https://mlsanity.com/paper/1510.00149/) (2015)
- [SNIP: Single-shot Network Pruning based on Connection Sensitivity](https://mlsanity.com/paper/1810.02340/) (2018)
- [Learning Structured Sparsity in Deep Neural Networks](https://mlsanity.com/paper/1608.03665/) (2016)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems](https://mlsanity.com/paper/1903.12394/) (2019)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Explainable Machine Learning for Scientific Insights and Discoveries](https://mlsanity.com/paper/1905.08883/) (2019)
- [A General Framework for Uncertainty Estimation in Deep Learning](https://mlsanity.com/paper/1907.06890/) (2019)
- [Bayesian Compression for Deep Learning](https://mlsanity.com/paper/1705.08665/) (2017)
- [The Case for Bayesian Deep Learning](https://mlsanity.com/paper/2001.10995/) (2020)
- [Symbolic Pregression: Discovering Physical Laws from Distorted Video](https://mlsanity.com/paper/2005.11212/) (2020)
- [Symbolic Regression Methods for Reinforcement Learning](https://mlsanity.com/paper/1903.09688/) (2019)
- [Learning Optimal Control of Synchronization in Networks of Coupled Oscillators using Genetic Programming-based Symbolic Regression](https://mlsanity.com/paper/1612.05276/) (2016)

## Built on by (most cited first, 3 of 3)

- [Toward Physically Plausible Data-Driven Models: A Novel Neural Network Approach to Symbolic Regression](https://mlsanity.com/paper/2302.00773/) (2023)
- [SymbolNet: Neural Symbolic Regression with Adaptive Dynamic Pruning for Compression](https://mlsanity.com/paper/2401.09949/) (2024)
- [Baseflow identification via explainable AI with Kolmogorov-Arnold networks](https://mlsanity.com/paper/2410.11587/) (2024)

## Abstract (arXiv)

Discovering governing equations from data is critical for diverse scientific disciplines as they can provide insights into the underlying phenomenon of dynamic systems. This work presents a new representation for governing equations by designing the Mathematical Operation Network (MathONet) with a deep neural network-like hierarchical structure. Specifically, the MathONet is stacked by several layers of unary operations (e.g., sin, cos, log) and binary operations (e.g., +,-), respectively. An initialized MathONet is typically regarded as a super-graph with a redundant structure, a sub-graph of which can yield the governing equation. We develop a sparse group Bayesian learning algorithm to extract the sub-graph by employing structurally constructed priors over the redundant mathematical operations. By demonstrating the chaotic Lorenz system, Lotka-Volterra system, and Kolmogorov-Petrovsky-Piskunov system, the proposed method can discover the ordinary differential equations (ODEs) and partial differential equations (PDEs) from the observations given limited mathematical operations, without any prior knowledge on possible expressions of the ODEs and PDEs.