# Beyond Accuracy and Complexity: The Effective Information Criterion for Structurally Stable Symbolic Regression

arXiv: https://arxiv.org/abs/2509.21780 · HTML: https://mlsanity.com/paper/2509.21780/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 30 references that are in the map.
- First posted on arXiv 26 Sep 2025.

## Builds on (most cited first, 21 of 21)

- [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)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [LLM-SR: Scientific Equation Discovery via Programming with Large Language Models](https://mlsanity.com/paper/2404.18400/) (2024)
- [Interpretable Scientific Discovery with Symbolic Regression: A Review](https://mlsanity.com/paper/2211.10873/) (2022)
- [Transformer-based Planning for Symbolic Regression](https://mlsanity.com/paper/2303.06833/) (2023)
- [LLM-SRBench: A New Benchmark for Scientific Equation Discovery with Large Language Models](https://mlsanity.com/paper/2504.10415/) (2025)
- [Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree search](https://mlsanity.com/paper/2205.13134/) (2022)
- [Deep symbolic regression for physics guided by units constraints: toward the automated discovery of physical laws](https://mlsanity.com/paper/2303.03192/) (2023)
- [SNIP: Bridging Mathematical Symbolic and Numeric Realms with Unified Pre-training](https://mlsanity.com/paper/2310.02227/) (2023)
- [Deep Generative Symbolic Regression with Monte-Carlo-Tree-Search](https://mlsanity.com/paper/2302.11223/) (2023)
- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [Symbolic regression via MDLformer-guided search: from minimizing prediction error to minimizing description length](https://mlsanity.com/paper/2411.03753/) (2024)
- [Call for Action: towards the next generation of symbolic regression benchmark](https://mlsanity.com/paper/2505.03977/) (2025)
- [A Neural Symbolic Model for Space Physics](https://mlsanity.com/paper/2503.07994/) (2025)
- [RSRM: Reinforcement Symbolic Regression Machine](https://mlsanity.com/paper/2305.14656/) (2023)
- [Automated discovery of symbolic laws governing skill acquisition from naturally occurring data](https://mlsanity.com/paper/2404.05689/) (2024)
- [Ab Initio Nonparametric Variable Selection for Scalable Symbolic Regression with Large p](https://mlsanity.com/paper/2410.13681/) (2024)

## Built on by (most cited first, 1 of 1)

- [Not Every Term Adds New Structure: Sobolev Novelty for Symbolic Regression](https://mlsanity.com/paper/2609.32597/) (2026)

## Abstract (arXiv)

Symbolic regression (SR) traditionally balances accuracy and complexity, implicitly assuming that simpler formulas are structurally more rational. We argue that this assumption is insufficient: existing algorithms often exploit this metric to discover accurate and compact but structurally irrational formulas that are numerically ill-conditioned and physically inexplicable. Inspired by the structural stability of real physical laws, we propose the Effective Information Criterion (EIC) to quantify formula rationality. EIC models formulas as information channels and measures the amplification of inherent rounding noise during recursive calculation, effectively distinguishing physically plausible structures from pathological ones without relying on ground truth. Our analysis reveals a stark structural stability gap between human-derived equations and SR-discovered results. By integrating EIC into SR workflows, we provide explicit structural guidance: for heuristic search, EIC steers algorithms toward stable regions to yield superior Pareto frontiers; for generative models, EIC-based filtering improves pre-training sample efficiency by 2-4 times and boosts generalization R2 by 22.4%. Finally, an extensive study with 108 human experts shows that EIC aligns with human preferences in 70% of cases, validating structural stability as a critical prerequisite for human-perceived interpretability.