# The Dynamics of Discoverability: How Trajectories and Priors Shape Equation Recovery

arXiv: https://arxiv.org/abs/2607.18490 · HTML: https://mlsanity.com/paper/2607.18490/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 35 references that are in the map.
- First posted on arXiv 20 Jul 2026.

## Builds on (most cited first, 16 of 16)

- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [LLM-SR: Scientific Equation Discovery via Programming with Large Language Models](https://mlsanity.com/paper/2404.18400/) (2024)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [PySINDy: A comprehensive Python package for robust sparse system identification](https://mlsanity.com/paper/2111.08481/) (2021)
- [Chaos as an interpretable benchmark for forecasting and data-driven modelling](https://mlsanity.com/paper/2110.05266/) (2021)
- [LLM-SRBench: A New Benchmark for Scientific Equation Discovery with Large Language Models](https://mlsanity.com/paper/2504.10415/) (2025)
- [ODEFormer: Symbolic Regression of Dynamical Systems with Transformers](https://mlsanity.com/paper/2310.05573/) (2023)
- [Extracting Sparse High-Dimensional Dynamics from Limited Data](https://mlsanity.com/paper/1707.08528/) (2017)
- [Rethinking Symbolic Regression Datasets and Benchmarks for Scientific Discovery](https://mlsanity.com/paper/2206.10540/) (2022)
- [Sparse Identification of Nonlinear Dynamical Systems via Reweighted ℓ1-regularized Least Squares](https://mlsanity.com/paper/2005.13232/) (2020)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [A toolkit for data-driven discovery of governing equations in high-noise regimes](https://mlsanity.com/paper/2111.04870/) (2021)
- [Benchmarking symbolic regression constant optimization schemes](https://mlsanity.com/paper/2412.02126/) (2024)
- [When is a System Discoverable from Data? Discovery Requires Chaos](https://mlsanity.com/paper/2511.08860/) (2025)
- [Invariant Measures in Time-Delay Coordinates for Unique Dynamical System Identification](https://mlsanity.com/paper/2412.00589/) (2024)
- [Information theory and discriminative sampling for model discovery](https://mlsanity.com/paper/2512.16000/) (2025)

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

- [Wrong Operator or Blind Design? A Post-Training Diagnostic for Physics-Informed Coefficient Learning](https://mlsanity.com/paper/2608.16925/) (2026)

## Abstract (arXiv)

How the dynamical regime of the observed system affects equation discovery has mainly been investigated through comparisons across systems. However, such comparisons vary both the equations and the dynamics, confounding the effect of the regime with the difficulty of recovering the equations symbolically. We separate the two by varying the forcing of Lorenz-84, moving its fully observed post-transient trajectories through fixed-point, periodic, and chaotic regimes while preserving the equations' functional form. Within each regime, we separately vary the amount of data, the noise, and prior knowledge of which terms the equations contain. We then measure how well two complementary approaches, sparse regression over a fixed library of candidate functions (SINDy) and an evolutionary search over symbolic expression trees (PySR), recover the true equations' terms and coefficients. We find that recovery depends on whether the sampled states distinguish combinations of candidate functions: equations remain poorly recovered from fixed-point data even when the candidate set contains only the true terms. We link the effect of the dynamics on both algorithms to one object: the moment matrix of the candidate functions under the invariant measure of the regime. Small eigenvalues mark weakly distinguishable combinations of candidate functions: we show that more data, less noise, and more prior knowledge can mitigate the resulting recovery difficulties, while a zero eigenvalue makes distinct equations indistinguishable on the visited states. Hence, a more precise prior needs less informative data, with consequences for data collection, method design, and evaluation.