# A computational framework for physics-informed symbolic regression with straightforward integration of domain knowledge

arXiv: https://arxiv.org/abs/2209.06257 · HTML: https://mlsanity.com/paper/2209.06257/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 128 references that are in the map.
- First posted on arXiv 13 Sep 2022.

## Builds on (most cited first, 23 of 23)

- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Probabilistic Model-Agnostic Meta-Learning](https://mlsanity.com/paper/1806.02817/) (2018)
- [Meta-Learning: A Survey](https://mlsanity.com/paper/1810.03548/) (2018)
- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Reinforcement learning based recommender systems: A survey](https://mlsanity.com/paper/2101.06286/) (2021)
- [AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity](https://mlsanity.com/paper/2006.10782/) (2020)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Synthetic Data Augmentation using GAN for Improved Liver Lesion Classification](https://mlsanity.com/paper/1801.02385/) (2018)
- [A Survey of Deep Meta-Learning](https://mlsanity.com/paper/2010.03522/) (2020)
- [PySINDy: A comprehensive Python package for robust sparse system identification](https://mlsanity.com/paper/2111.08481/) (2021)
- [Symbolic Regression is NP-hard](https://mlsanity.com/paper/2207.01018/) (2022)
- [Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks](https://mlsanity.com/paper/1905.01205/) (2019)
- [Visualizing the Feature Importance for Black Box Models](https://mlsanity.com/paper/1804.06620/) (2018)
- [Where are we now? A large benchmark study of recent symbolic regression methods](https://mlsanity.com/paper/1804.09331/) (2018)
- [Monotonic Calibrated Interpolated Look-Up Tables](https://mlsanity.com/paper/1505.06378/) (2015)
- [Shape-constrained Symbolic Regression -- Improving Extrapolation with Prior Knowledge](https://mlsanity.com/paper/2103.15624/) (2021)
- [Symbolic Regression by Exhaustive Search: Reducing the Search Space Using Syntactical Constraints and Efficient Semantic Structure Deduplication](https://mlsanity.com/paper/2109.13895/) (2021)
- [Elite Bases Regression: A Real-time Algorithm for Symbolic Regression](https://mlsanity.com/paper/1704.07313/) (2017)
- [A Probabilistic Linear Genetic Programming with Stochastic Context-Free Grammar for solving Symbolic Regression problems](https://mlsanity.com/paper/1704.00828/) (2017)
- [Accelerating Genetic Programming using GPUs](https://mlsanity.com/paper/2110.11226/) (2021)
- [On Enhancing Genetic Algorithms Using New Crossovers](https://mlsanity.com/paper/1801.02335/) (2018)

## Built on by (most cited first, 30 of 31)

- [Deep symbolic regression for physics guided by units constraints: toward the automated discovery of physical laws](https://mlsanity.com/paper/2303.03192/) (2023)
- [SymbolNet: Neural Symbolic Regression with Adaptive Dynamic Pruning for Compression](https://mlsanity.com/paper/2401.09949/) (2024)
- [Knowledge Integration for Physics-informed Symbolic Regression Using Pre-trained Large Language Models](https://mlsanity.com/paper/2509.03036/) (2025)
- [AI-Assisted Discovery of Quantitative and Formal Models in Social Science](https://mlsanity.com/paper/2210.00563/) (2022)
- [Vertical Symbolic Regression via Deep Policy Gradient](https://mlsanity.com/paper/2402.00254/) (2024)
- [Symbolic Regression as Feature Engineering Method for Machine and Deep Learning Regression Tasks](https://mlsanity.com/paper/2311.06028/) (2023)
- [A Comparison of Recent Algorithms for Symbolic Regression to Genetic Programming](https://mlsanity.com/paper/2406.03585/) (2024)
- [Scaling Up Unbiased Search-based Symbolic Regression](https://mlsanity.com/paper/2506.19626/) (2025)
- [Unit-Aware Genetic Programming for the Development of Empirical Equations](https://mlsanity.com/paper/2405.18896/) (2024)
- [Active Symbolic Discovery of Ordinary Differential Equations via Phase Portrait Sketching](https://mlsanity.com/paper/2409.01416/) (2024)
- [Ab Initio Nonparametric Variable Selection for Scalable Symbolic Regression with Large p](https://mlsanity.com/paper/2410.13681/) (2024)
- [SymbolFit: Automatic Parametric Modeling with Symbolic Regression](https://mlsanity.com/paper/2411.09851/) (2024)
- [Incorporating Background Knowledge in Symbolic Regression using a Computer Algebra System](https://mlsanity.com/paper/2301.11919/) (2023)
- [Exploration-Exploitation Model of Moth-Inspired Olfactory Navigation](https://mlsanity.com/paper/2312.11492/) (2023)
- [ISR: Invertible Symbolic Regression](https://mlsanity.com/paper/2405.06848/) (2024)
- [Sparse Interpretable Deep Learning with LIES Networks for Symbolic Regression](https://mlsanity.com/paper/2506.08267/) (2025)
- [EGG-SR: Embedding Symbolic Equivalence into Symbolic Regression via Equality Graph](https://mlsanity.com/paper/2511.05849/) (2025)
- [Beyond empirical models: Discovering new constitutive laws in solids with graph-based equation discovery](https://mlsanity.com/paper/2511.09906/) (2025)
- [Vertical Symbolic Regression](https://mlsanity.com/paper/2312.11955/) (2023)
- [Discovering Nuclear Models from Symbolic Machine Learning](https://mlsanity.com/paper/2404.11477/) (2024)
- [A Personalised Learning Tool for Physics Undergraduate Students Built On a Large Language Model for Symbolic Regression](https://mlsanity.com/paper/2407.00065/) (2024)
- [Introducing 'Inside' Out of Distribution](https://mlsanity.com/paper/2407.04534/) (2024)
- [Enhancing Symbolic Regression and Universal Physics-Informed Neural Networks with Dimensional Analysis](https://mlsanity.com/paper/2411.15919/) (2024)
- [Pulling the Carpet Below the Learner's Feet: Genetic Algorithm To Learn Ensemble Machine Learning Model During Concept Drift](https://mlsanity.com/paper/2412.09035/) (2024)
- [SPINEX_ Symbolic Regression: Similarity-based Symbolic Regression with Explainable Neighbors Exploration](https://mlsanity.com/paper/2411.03358/) (2024)
- [Augmented Physics-Based Li-ion Battery Model via Adaptive Ensemble Sparse Learning and Conformal Prediction](https://mlsanity.com/paper/2507.00353/) (2025)
- [Gaussian Process Regression -- Neural Network Hybrid with Optimized Redundant Coordinates](https://mlsanity.com/paper/2509.08457/) (2025)
- [Data Enrichment for Symbolic Regression Using Diffusion Models](https://mlsanity.com/paper/2606.00988/) (2026)
- [Discovering Multiscale Deep Formulas in Complex Systems via Neural-Guided Lambda Calculus](https://mlsanity.com/paper/2606.07426/) (2026)
- [Automatic Ordinary Differential Equations Discovery For Biological Systems Using Large Language Model Powered Agentic System](https://mlsanity.com/paper/2607.13608/) (2026)

## Abstract (arXiv)

Discovering a meaningful symbolic expression that explains experimental data is a fundamental challenge in many scientific fields. We present a novel, open-source computational framework called Scientist-Machine Equation Detector (SciMED), which integrates scientific discipline wisdom in a scientist-in-the-loop approach, with state-of-the-art symbolic regression (SR) methods. SciMED combines a wrapper selection method, that is based on a genetic algorithm, with automatic machine learning and two levels of SR methods. We test SciMED on five configurations of a settling sphere, with and without aerodynamic non-linear drag force, and with excessive noise in the measurements. We show that SciMED is sufficiently robust to discover the correct physically meaningful symbolic expressions from the data, and demonstrate how the integration of domain knowledge enhances its performance. Our results indicate better performance on these tasks than the state-of-the-art SR software packages , even in cases where no knowledge is integrated. Moreover, we demonstrate how SciMED can alert the user about possible missing features, unlike the majority of current SR systems.