# A PINN Approach to Symbolic Differential Operator Discovery with Sparse Data

arXiv: https://arxiv.org/abs/2212.04630 · HTML: https://mlsanity.com/paper/2212.04630/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 27 references that are in the map.
- First posted on arXiv 9 Dec 2022.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [SymbolicGPT: A Generative Transformer Model for Symbolic Regression](https://mlsanity.com/paper/2106.14131/) (2021)
- [Transfer learning based multi-fidelity physics informed deep neural network](https://mlsanity.com/paper/2005.10614/) (2020)
- [DAE-PINN: A Physics-Informed Neural Network Model for Simulating Differential-Algebraic Equations with Application to Power Networks](https://mlsanity.com/paper/2109.04304/) (2021)
- [Deep learning prediction of patient response time course from early data via neural-pharmacokinetic/pharmacodynamic modeling](https://mlsanity.com/paper/2010.11769/) (2020)

## Built on by (most cited first, 6 of 6)

- [Correcting model misspecification in physics-informed neural networks (PINNs)](https://mlsanity.com/paper/2310.10776/) (2023)
- [Discovering a reaction-diffusion model for Alzheimer's disease by combining PINNs with symbolic regression](https://mlsanity.com/paper/2307.08107/) (2023)
- [Neuro-Symbolic AI: Explainability, Challenges, and Future Trends](https://mlsanity.com/paper/2411.04383/) (2024)
- [Learning Chemotherapy Drug Action via Universal Physics-Informed Neural Networks](https://mlsanity.com/paper/2404.08019/) (2024)
- [Deep learning for model correction of dynamical systems with data scarcity](https://mlsanity.com/paper/2410.17913/) (2024)
- [Bridging the Gap: Representation Spaces in Neuro-Symbolic AI](https://mlsanity.com/paper/2411.04393/) (2024)

## Abstract (arXiv)

Given ample experimental data from a system governed by differential equations, it is possible to use deep learning techniques to construct the underlying differential operators. In this work we perform symbolic discovery of differential operators in a situation where there is sparse experimental data. This small data regime in machine learning can be made tractable by providing our algorithms with prior information about the underlying dynamics. Physics Informed Neural Networks (PINNs) have been very successful in this regime (reconstructing entire ODE solutions using only a single point or entire PDE solutions with very few measurements of the initial condition). We modify the PINN approach by adding a neural network that learns a representation of unknown hidden terms in the differential equation. The algorithm yields both a surrogate solution to the differential equation and a black-box representation of the hidden terms. These hidden term neural networks can then be converted into symbolic equations using symbolic regression techniques like AI Feynman. In order to achieve convergence of these neural networks, we provide our algorithms with (noisy) measurements of both the initial condition as well as (synthetic) experimental data obtained at later times. We demonstrate strong performance of this approach even when provided with very few measurements of noisy data in both the ODE and PDE regime.