# MDBench: Benchmarking Data-Driven Methods for Model Discovery

arXiv: https://arxiv.org/abs/2509.20529 · HTML: https://mlsanity.com/paper/2509.20529/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 58 references that are in the map.
- First posted on arXiv 24 Sep 2025.

## Builds on (most cited first, 22 of 22)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (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)
- [Learning Equations for Extrapolation and Control](https://mlsanity.com/paper/1806.07259/) (2018)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Language Model Crossover: Variation through Few-Shot Prompting](https://mlsanity.com/paper/2302.12170/) (2023)
- [Extrapolation and learning equations](https://mlsanity.com/paper/1610.02995/) (2016)
- [Integration of Neural Network-Based Symbolic Regression in Deep Learning for Scientific Discovery](https://mlsanity.com/paper/1912.04825/) (2019)
- [Chaos as an interpretable benchmark for forecasting and data-driven modelling](https://mlsanity.com/paper/2110.05266/) (2021)
- [Transformer-based Planning for Symbolic Regression](https://mlsanity.com/paper/2303.06833/) (2023)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [ODEFormer: Symbolic Regression of Dynamical Systems with Transformers](https://mlsanity.com/paper/2310.05573/) (2023)
- [CFDBench: A Large-Scale Benchmark for Machine Learning Methods in Fluid Dynamics](https://mlsanity.com/paper/2310.05963/) (2023)
- [In-Context Symbolic Regression: Leveraging Large Language Models for Function Discovery](https://mlsanity.com/paper/2404.19094/) (2024)
- [An Extensible Benchmark Suite for Learning to Simulate Physical Systems](https://mlsanity.com/paper/2108.07799/) (2021)
- [A Bayesian Framework for learning governing Partial Differential Equation from Data](https://mlsanity.com/paper/2306.04894/) (2023)
- [A Bayesian Approach for Spatio-Temporal Data-Driven Dynamic Equation Discovery](https://mlsanity.com/paper/2209.02750/) (2022)

## Built on by (most cited first, 7 of 7)

- [LLM-ACES: Closed-Loop Discovery of Dynamical Systems with LLM-Guided Adaptive Search](https://mlsanity.com/paper/2606.25039/) (2026)
- [LLM-ODE: Data-driven Discovery of Dynamical Systems with Large Language Models](https://mlsanity.com/paper/2603.20910/) (2026)
- [Data-driven discovery of governing differential equations across physical systems](https://mlsanity.com/paper/2606.09638/) (2026)
- [When Does AI for PDEs Yield Scientific Evidence?](https://mlsanity.com/paper/2608.22504/) (2026)
- [Automatic Ordinary Differential Equations Discovery For Biological Systems Using Large Language Model Powered Agentic System](https://mlsanity.com/paper/2607.13608/) (2026)
- [On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancement](https://mlsanity.com/paper/2607.23753/) (2026)
- [STRIDE: State-Transition Representation via Increment Dynamics and Evolution](https://mlsanity.com/paper/2609.32367/) (2026)

## Abstract (arXiv)

Model discovery aims to uncover governing differential equations of dynamical systems directly from experimental data. Benchmarking such methods is essential for tracking progress and understanding trade-offs in the field. While prior efforts have focused mostly on identifying single equations, typically framed as symbolic regression, there remains a lack of comprehensive benchmarks for discovering dynamical models. To address this, we introduce MDBench, an open-source benchmarking framework for evaluating model discovery methods on dynamical systems. MDBench assesses 12 algorithms on 14 partial differential equations (PDEs) and 63 ordinary differential equations (ODEs) under varying levels of noise. Evaluation metrics include derivative prediction accuracy, model complexity, and equation fidelity. We also introduce seven challenging PDE systems from fluid dynamics and thermodynamics, revealing key limitations in current methods. Our findings illustrate that linear methods and genetic programming methods achieve the lowest prediction error for PDEs and ODEs, respectively. Moreover, linear models are in general more robust against noise. MDBench accelerates the advancement of model discovery methods by offering a rigorous, extensible benchmarking framework and a rich, diverse collection of dynamical system datasets, enabling systematic evaluation, comparison, and improvement of equation accuracy and robustness.