# Symbolic Regression Driven by Training Data and Prior Knowledge

arXiv: https://arxiv.org/abs/2004.11947 · HTML: https://mlsanity.com/paper/2004.11947/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Apr 2020.

## Built on by (most cited first, 13 of 13)

- [SymFormer: End-to-end symbolic regression using transformer-based architecture](https://mlsanity.com/paper/2205.15764/) (2022)
- [Evolving Scientific Discovery by Unifying Data and Background Knowledge with AI Hilbert](https://mlsanity.com/paper/2308.09474/) (2023)
- [Toward Physically Plausible Data-Driven Models: A Novel Neural Network Approach to Symbolic Regression](https://mlsanity.com/paper/2302.00773/) (2023)
- [AI Descartes: Combining Data and Theory for Derivable Scientific Discovery](https://mlsanity.com/paper/2109.01634/) (2021)
- [The Need for Verification in AI-Driven Scientific Discovery](https://mlsanity.com/paper/2509.01398/) (2025)
- [A Comparison of Recent Algorithms for Symbolic Regression to Genetic Programming](https://mlsanity.com/paper/2406.03585/) (2024)
- [Shape-constrained Symbolic Regression with NSGA-III](https://mlsanity.com/paper/2209.13851/) (2022)
- [Shape Constraints in Symbolic Regression using Penalized Least Squares](https://mlsanity.com/paper/2405.20800/) (2024)
- [ViSymRe: Vision Multimodal Symbolic Regression](https://mlsanity.com/paper/2412.11139/) (2024)
- [Beyond empirical models: Discovering new constitutive laws in solids with graph-based equation discovery](https://mlsanity.com/paper/2511.09906/) (2025)
- [Neuro-Evolutionary Approach to Physics-Aware Symbolic Regression](https://mlsanity.com/paper/2504.16503/) (2025)
- [Bridging the Gap Between Scientific Laws Derived by AI Systems and Canonical Knowledge via Abductive Inference with AI-Noether](https://mlsanity.com/paper/2509.23004/) (2025)
- [AutoSR: Automatic Symbolic Regression by Searching Research States](https://mlsanity.com/paper/2608.16876/) (2026)

## Abstract (arXiv)

In symbolic regression, the search for analytic models is typically driven purely by the prediction error observed on the training data samples. However, when the data samples do not sufficiently cover the input space, the prediction error does not provide sufficient guidance toward desired models. Standard symbolic regression techniques then yield models that are partially incorrect, for instance, in terms of their steady-state characteristics or local behavior. If these properties were considered already during the search process, more accurate and relevant models could be produced. We propose a multi-objective symbolic regression approach that is driven by both the training data and the prior knowledge of the properties the desired model should manifest. The properties given in the form of formal constraints are internally represented by a set of discrete data samples on which candidate models are exactly checked. The proposed approach was experimentally evaluated on three test problems with results clearly demonstrating its capability to evolve realistic models that fit the training data well while complying with the prior knowledge of the desired model characteristics at the same time. It outperforms standard symbolic regression by several orders of magnitude in terms of the mean squared deviation from a reference model.