# AI-Lorenz: A physics-data-driven framework for black-box and gray-box identification of chaotic systems with symbolic regression

arXiv: https://arxiv.org/abs/2312.14237 · HTML: https://mlsanity.com/paper/2312.14237/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 63 references that are in the map.
- First posted on arXiv 21 Dec 2023.

## Builds on (most cited first, 19 of 19)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting](https://mlsanity.com/paper/2010.04456/) (2020)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders](https://mlsanity.com/paper/2201.05136/) (2022)
- [Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl](https://mlsanity.com/paper/2305.01582/) (2023)
- [Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems](https://mlsanity.com/paper/2203.15706/) (2022)
- [Discovery of Nonlinear Dynamical Systems using a Runge-Kutta Inspired Dictionary-based Sparse Regression Approach](https://mlsanity.com/paper/2105.04869/) (2021)
- [AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identification](https://mlsanity.com/paper/2310.01433/) (2023)
- [Structure-preserving Sparse Identification of Nonlinear Dynamics for Data-driven Modeling](https://mlsanity.com/paper/2109.05364/) (2021)
- [Interpretable Polynomial Neural Ordinary Differential Equations](https://mlsanity.com/paper/2208.05072/) (2022)
- [Bayesian System ID: Optimal management of parameter, model, and measurement uncertainty](https://mlsanity.com/paper/2003.02359/) (2020)
- [The Theory of Connections. Connecting Points](https://mlsanity.com/paper/1702.06862/) (2017)
- [Sparse dynamical system identification with simultaneous structural parameters and initial condition estimation](https://mlsanity.com/paper/2204.10472/) (2022)
- [Implementation and (Inverse Modified) Error Analysis for implicitly-templated ODE-nets](https://mlsanity.com/paper/2303.17824/) (2023)
- [Constraint-Based Visual Generation](https://mlsanity.com/paper/1807.09202/) (2018)
- [Some of the variables, some of the parameters, some of the times, with some physics known: Identification with partial information](https://mlsanity.com/paper/2304.14214/) (2023)
- [Data-driven and Physics Informed Modelling of Chinese Hamster Ovary Cell Bioreactors](https://mlsanity.com/paper/2305.03257/) (2023)

## Built on by (most cited first, 9 of 9)

- [Integrating Neural Operators with Diffusion Models Improves Spectral Representation in Turbulence Modeling](https://mlsanity.com/paper/2409.08477/) (2024)
- [FMint: Bridging Human Designed and Data Pretrained Models for Differential Equation Foundation Model](https://mlsanity.com/paper/2404.14688/) (2024)
- [A Neural Symbolic Model for Space Physics](https://mlsanity.com/paper/2503.07994/) (2025)
- [Improving the Adaptive Moment Estimation (ADAM) stochastic optimizer through an Implicit-Explicit (IMEX) time-stepping approach](https://mlsanity.com/paper/2403.13704/) (2024)
- [Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiology](https://mlsanity.com/paper/2408.07201/) (2024)
- [Prior-Guided Symbolic Regression: Towards Scientific Consistency in Equation Discovery](https://mlsanity.com/paper/2602.13021/) (2026)
- [Connecting the geometry and dynamics of many-body complex systems with message passing neural operators](https://mlsanity.com/paper/2502.15913/) (2025)
- [Learning task-specific predictive models for scientific computing](https://mlsanity.com/paper/2506.03835/) (2025)
- [Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis](https://mlsanity.com/paper/2609.04329/) (2026)

## Abstract (arXiv)

Discovering mathematical models that characterize the observed behavior of dynamical systems remains a major challenge, especially for systems in a chaotic regime. The challenge is even greater when the physics underlying such systems is not yet understood, and scientific inquiry must solely rely on empirical data. Driven by the need to fill this gap, we develop a framework that learns mathematical expressions modeling complex dynamical behaviors by identifying differential equations from noisy and sparse observable data. We train a small neural network to learn the dynamics of a system, its rate of change in time, and missing model terms, which are used as input for a symbolic regression algorithm to autonomously distill the explicit mathematical terms. This, in turn, enables us to predict the future evolution of the dynamical behavior. The performance of this framework is validated by recovering the right-hand sides and unknown terms of certain complex, chaotic systems such as the well-known Lorenz system, a six-dimensional hyperchaotic system, and the non-autonomous Sprott chaotic system, and comparing them with their known analytical expressions.