# AI Descartes: Combining Data and Theory for Derivable Scientific Discovery arXiv: https://arxiv.org/abs/2109.01634 ยท HTML: https://mlsanity.com/paper/2109.01634/ ## Key facts - Cited by 8 papers in the map (data through 14 Oct 2026). - Builds on 32 of its 87 references that are in the map. - First posted on arXiv 3 Sep 2021. ## Builds on (most cited first, 30 of 32) - [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019) - [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020) - [A Semantic Loss Function for Deep Learning with Symbolic Knowledge](https://mlsanity.com/paper/1711.11157/) (2017) - [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019) - [Learning Explanatory Rules from Noisy Data](https://mlsanity.com/paper/1711.04574/) (2017) - [Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems](https://mlsanity.com/paper/1903.12394/) (2019) - [AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity](https://mlsanity.com/paper/2006.10782/) (2020) - [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018) - [Extrapolation and learning equations](https://mlsanity.com/paper/1610.02995/) (2016) - [Neuro-Symbolic Program Synthesis](https://mlsanity.com/paper/1611.01855/) (2016) - [Bayesian Symbolic Regression](https://mlsanity.com/paper/1910.08892/) (2019) - [Learning Compositional Rules via Neural Program Synthesis](https://mlsanity.com/paper/2003.05562/) (2020) - [A Logic-Driven Framework for Consistency of Neural Models](https://mlsanity.com/paper/1909.00126/) (2019) - [A Bayesian machine scientist to aid in the solution of challenging scientific problems](https://mlsanity.com/paper/2004.12157/) (2020) - [Incorporating Domain Knowledge into Deep Neural Networks](https://mlsanity.com/paper/2103.00180/) (2021) - [Augmenting Neural Networks with First-order Logic](https://mlsanity.com/paper/1906.06298/) (2019) - [Embedding Symbolic Knowledge into Deep Networks](https://mlsanity.com/paper/1909.01161/) (2019) - [Inductive Learning of Answer Set Programs from Noisy Examples](https://mlsanity.com/paper/1808.08441/) (2018) - [Mathematical Reasoning in Latent Space](https://mlsanity.com/paper/1909.11851/) (2019) - [Integrating Deep Learning with Logic Fusion for Information Extraction](https://mlsanity.com/paper/1912.03041/) (2019) - [Symbolic Regression Driven by Training Data and Prior Knowledge](https://mlsanity.com/paper/2004.11947/) (2020) - [Sum-of-squares proofs and the quest toward optimal algorithms](https://mlsanity.com/paper/1404.5236/) (2014) - [Learning dynamic polynomial proofs](https://mlsanity.com/paper/1906.01681/) (2019) - [On Universal Approximation by Neural Networks with Uniform Guarantees on Approximation of Infinite Dimensional Maps](https://mlsanity.com/paper/1910.01545/) (2019) - [Differentiable Genetic Programming](https://mlsanity.com/paper/1611.04766/) (2016) - [Tensor network approaches for learning non-linear dynamical laws](https://mlsanity.com/paper/2002.12388/) (2020) - [Globally Optimal Symbolic Regression](https://mlsanity.com/paper/1710.10720/) (2017) - [Logic Guided Genetic Algorithms](https://mlsanity.com/paper/2010.11328/) (2020) - [Constraint-Based Visual Generation](https://mlsanity.com/paper/1807.09202/) (2018) - [A Deep Reinforcement Learning Approach to First-Order Logic Theorem Proving](https://mlsanity.com/paper/1911.02065/) (2019) ## Built on by (most cited first, 8 of 8) - [Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl](https://mlsanity.com/paper/2305.01582/) (2023) - [A Triumvirate of AI Driven Theoretical Discovery](https://mlsanity.com/paper/2405.19973/) (2024) - [Incorporating Background Knowledge in Symbolic Regression using a Computer Algebra System](https://mlsanity.com/paper/2301.11919/) (2023) - [Finite Element Analysis and Machine Learning Guided Design of Carbon Fiber Organosheet-based Battery Enclosures for Crashworthiness](https://mlsanity.com/paper/2309.00637/) (2023) - [Bayesian Experimental Design for Symbolic Discovery](https://mlsanity.com/paper/2211.15860/) (2022) - [Symbolic Classification-Enabled LHC Limits for BSM Global Fits](https://mlsanity.com/paper/2605.22330/) (2026) - [How AI Agents Discover Scientific Equations: From Hydrotope Rediscovery to New Water-Wave Amplitudes](https://mlsanity.com/paper/2610.00435/) (2026) - [The Future of AI-assisted Physics: Discovering, Verifying and Organizing Results via Formalization](https://mlsanity.com/paper/2610.14059/) (2026) ## Abstract (arXiv) Scientists have long aimed to discover meaningful formulae which accurately describe experimental data. A common approach is to manually create mathematical models of natural phenomena using domain knowledge, and then fit these models to data. In contrast, machine-learning algorithms automate the construction of accurate data-driven models while consuming large amounts of data. The problem of incorporating prior knowledge in the form of constraints on the functional form of a learned model (e.g., nonnegativity) has been explored in the literature. However, finding models that are consistent with prior knowledge expressed in the form of general logical axioms (e.g., conservation of energy) is an open problem. We develop a method to enable principled derivations of models of natural phenomena from axiomatic knowledge and experimental data by combining logical reasoning with symbolic regression. We demonstrate these concepts for Kepler's third law of planetary motion, Einstein's relativistic time-dilation law, and Langmuir's theory of adsorption, automatically connecting experimental data with background theory in each case. We show that laws can be discovered from few data points when using formal logical reasoning to distinguish the correct formula from a set of plausible formulas that have similar error on the data. The combination of reasoning with machine learning provides generalizeable insights into key aspects of natural phenomena. We envision that this combination will enable derivable discovery of fundamental laws of science and believe that our work is an important step towards automating the scientific method.