# Discovering Causal Relations and Equations from Data

arXiv: https://arxiv.org/abs/2305.13341 · HTML: https://mlsanity.com/paper/2305.13341/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 92 of its 439 references that are in the map.
- First posted on arXiv 21 May 2023.

## Builds on (most cited first, 30 of 92)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Learning Sparse Neural Networks through L0 Regularization](https://mlsanity.com/paper/1712.01312/) (2017)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Causal inference using invariant prediction: identification and confidence intervals](https://mlsanity.com/paper/1501.01332/) (2015)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Detecting causal associations in large nonlinear time series datasets](https://mlsanity.com/paper/1702.07007/) (2017)
- [Gradient-Based Neural DAG Learning](https://mlsanity.com/paper/1906.02226/) (2019)
- [Differentiable Causal Discovery from Interventional Data](https://mlsanity.com/paper/2007.01754/) (2020)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Learning Sparse Nonparametric DAGs](https://mlsanity.com/paper/1909.13189/) (2019)
- [DYNOTEARS: Structure Learning from Time-Series Data](https://mlsanity.com/paper/2002.00498/) (2020)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [On the Role of Sparsity and DAG Constraints for Learning Linear DAGs](https://mlsanity.com/paper/2006.10201/) (2020)
- [Causal Inference and Causal Explanation with Background Knowledge](https://mlsanity.com/paper/1302.4972/) (2013)
- [Causal Inference in the Presence of Latent Variables and Selection Bias](https://mlsanity.com/paper/1302.4983/) (2013)
- [Causal Discovery from Heterogeneous/Nonstationary Data with Independent Changes](https://mlsanity.com/paper/1903.01672/) (2019)
- [Flexible statistical inference for mechanistic models of neural dynamics](https://mlsanity.com/paper/1711.01861/) (2017)
- [Beware of the Simulated DAG! Causal Discovery Benchmarks May Be Easy To Game](https://mlsanity.com/paper/2102.13647/) (2021)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Learning Equations for Extrapolation and Control](https://mlsanity.com/paper/1806.07259/) (2018)
- [AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity](https://mlsanity.com/paper/2006.10782/) (2020)
- [Invariant Causal Prediction for Nonlinear Models](https://mlsanity.com/paper/1706.08576/) (2017)
- [DAGMA: Learning DAGs via M-matrices and a Log-Determinant Acyclicity Characterization](https://mlsanity.com/paper/2209.08037/) (2022)
- [Discovering contemporaneous and lagged causal relations in autocorrelated nonlinear time series datasets](https://mlsanity.com/paper/2003.03685/) (2020)
- [Foundations of Structural Causal Models with Cycles and Latent Variables](https://mlsanity.com/paper/1611.06221/) (2016)
- [Integration of Neural Network-Based Symbolic Regression in Deep Learning for Scientific Discovery](https://mlsanity.com/paper/1912.04825/) (2019)

## Built on by (most cited first, 30 of 36)

- [Large Language Models for Constrained-Based Causal Discovery](https://mlsanity.com/paper/2406.07378/) (2024)
- [Do AI models produce better weather forecasts than physics-based models? A quantitative evaluation case study of Storm Ciar'an](https://mlsanity.com/paper/2312.02658/) (2023)
- [ODE Discovery for Longitudinal Heterogeneous Treatment Effects Inference](https://mlsanity.com/paper/2403.10766/) (2024)
- [Equilibrium in the Computing Continuum through Active Inference](https://mlsanity.com/paper/2311.16769/) (2023)
- [Unitless Unrestricted Markov-Consistent SCM Generation: Better Benchmark Datasets for Causal Discovery](https://mlsanity.com/paper/2503.17037/) (2025)
- [Causal Graph Neural Networks for Wildfire Danger Prediction](https://mlsanity.com/paper/2403.08414/) (2024)
- [AI-Newton: A Concept-Driven Physical Law Discovery System without Prior Physical Knowledge](https://mlsanity.com/paper/2504.01538/) (2025)
- [Explainable AI: Learning from the Learners](https://mlsanity.com/paper/2601.05525/) (2026)
- [Response Theory via Generative Score Modeling](https://mlsanity.com/paper/2402.01029/) (2024)
- [Decoding complexity: how machine learning is redefining scientific discovery](https://mlsanity.com/paper/2405.04161/) (2024)
- [AI for Extreme Event Modeling and Understanding: Methodologies and Challenges](https://mlsanity.com/paper/2406.20080/) (2024)
- [Earth System Data Cubes: Avenues for advancing Earth system research](https://mlsanity.com/paper/2408.02348/) (2024)
- [Analyzing Generalization in Pre-Trained Symbolic Regression](https://mlsanity.com/paper/2509.19849/) (2025)
- [On the choice of training data for machine learning of geostrophic mesoscale turbulence](https://mlsanity.com/paper/2307.00734/) (2023)
- [An analysis of optimization problems involving ReLU neural networks](https://mlsanity.com/paper/2502.03016/) (2025)
- [Deep Koopman operator framework for causal discovery in nonlinear dynamical systems](https://mlsanity.com/paper/2505.14828/) (2025)
- [Discovering equations from data: symbolic regression in dynamical systems](https://mlsanity.com/paper/2508.20257/) (2025)
- [Electron neural closure for turbulent magnetosheath simulations: energy channels](https://mlsanity.com/paper/2510.00282/) (2025)
- [Physics-Informed Neural Networks for Biological 2D+t Reaction-Diffusion Systems](https://mlsanity.com/paper/2604.18548/) (2026)
- [Discovery of Nonlinear Dynamics with Automated Basis Function Generation](https://mlsanity.com/paper/2605.09696/) (2026)
- [Discovering Dynamic Effective Connectome of Brain with Bayesian Dynamic DAG Learning](https://mlsanity.com/paper/2309.07080/) (2023)
- [Machine-learning invariant foliations in forced systems for reduced order modelling](https://mlsanity.com/paper/2403.14514/) (2024)
- [Quantum-enhanced causal discovery for a small number of samples](https://mlsanity.com/paper/2501.05007/) (2025)
- [Discovering Generalizable Governing Equations for Graph Dynamical Systems with Interpretable Neural Networks](https://mlsanity.com/paper/2508.18173/) (2025)
- [Beyond Accuracy and Complexity: The Effective Information Criterion for Structurally Stable Symbolic Regression](https://mlsanity.com/paper/2509.21780/) (2025)
- [The Machine Learning Approach to Moment Closure Relations for Plasma: A Review](https://mlsanity.com/paper/2511.22486/) (2025)
- [Projecting infinite time series graphs to finite marginal graphs using number theory](https://mlsanity.com/paper/2310.05526/) (2023)
- [Causal machine learning for sustainable agroecosystems](https://mlsanity.com/paper/2408.13155/) (2024)
- [Discovery and inversion of the viscoelastic wave equation in inhomogeneous media](https://mlsanity.com/paper/2409.18370/) (2024)
- [MXMap: A Multivariate Cross Mapping Framework for Causal Discovery in Dynamical Systems](https://mlsanity.com/paper/2502.03802/) (2025)

## Abstract (arXiv)

Physics is a field of science that has traditionally used the scientific method to answer questions about why natural phenomena occur and to make testable models that explain the phenomena. Discovering equations, laws and principles that are invariant, robust and causal explanations of the world has been fundamental in physical sciences throughout the centuries. Discoveries emerge from observing the world and, when possible, performing interventional studies in the system under study. With the advent of big data and the use of data-driven methods, causal and equation discovery fields have grown and made progress in computer science, physics, statistics, philosophy, and many applied fields. All these domains are intertwined and can be used to discover causal relations, physical laws, and equations from observational data. This paper reviews the concepts, methods, and relevant works on causal and equation discovery in the broad field of Physics and outlines the most important challenges and promising future lines of research. We also provide a taxonomy for observational causal and equation discovery, point out connections, and showcase a complete set of case studies in Earth and climate sciences, fluid dynamics and mechanics, and the neurosciences. This review demonstrates that discovering fundamental laws and causal relations by observing natural phenomena is being revolutionised with the efficient exploitation of observational data, modern machine learning algorithms and the interaction with domain knowledge. Exciting times are ahead with many challenges and opportunities to improve our understanding of complex systems.