# AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identification

arXiv: https://arxiv.org/abs/2310.01433 · HTML: https://mlsanity.com/paper/2310.01433/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 69 references that are in the map.
- First posted on arXiv 29 Sep 2023.

## Builds on (most cited first, 23 of 23)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Symbolic Regression is NP-hard](https://mlsanity.com/paper/2207.01018/) (2022)
- [Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl](https://mlsanity.com/paper/2305.01582/) (2023)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [Optimal control of PDEs using physics-informed neural networks](https://mlsanity.com/paper/2111.09880/) (2021)
- [Deep Kronecker neural networks: A general framework for neural networks with adaptive activation functions](https://mlsanity.com/paper/2105.09513/) (2021)
- [Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approach](https://mlsanity.com/paper/2201.13323/) (2022)
- [Discovering a reaction-diffusion model for Alzheimer's disease by combining PINNs with symbolic regression](https://mlsanity.com/paper/2307.08107/) (2023)
- [Systems Biology: Identifiability analysis and parameter identification via systems-biology informed neural networks](https://mlsanity.com/paper/2202.01723/) (2022)
- [An Approach to Symbolic Regression Using Feyn](https://mlsanity.com/paper/2104.05417/) (2021)
- [Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEs](https://mlsanity.com/paper/2203.05337/) (2022)
- [Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data](https://mlsanity.com/paper/2205.13545/) (2022)
- [A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Data](https://mlsanity.com/paper/2305.10706/) (2023)
- [A Greedy Search Tree Heuristic for Symbolic Regression](https://mlsanity.com/paper/1801.01807/) (2018)
- [Symbolic regression outperforms other models for small data sets](https://mlsanity.com/paper/2103.15147/) (2021)
- [The Theory of Connections. Connecting Points](https://mlsanity.com/paper/1702.06862/) (2017)
- [Partial observations and conservation laws: Grey-box modeling in biotechnology and optogenetics](https://mlsanity.com/paper/1909.04234/) (2019)
- [Physics-informed neural networks for predicting gas flow dynamics and unknown parameters in diesel engines](https://mlsanity.com/paper/2304.13799/) (2023)
- [Black and Gray Box Learning of Amplitude Equations: Application to Phase Field Systems](https://mlsanity.com/paper/2207.03954/) (2022)
- [Splitting physics-informed neural networks for inferring the dynamics of integer- and fractional-order neuron models](https://mlsanity.com/paper/2304.13205/) (2023)

## Built on by (most cited first, 29 of 29)

- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Integrating Neural Operators with Diffusion Models Improves Spectral Representation in Turbulence Modeling](https://mlsanity.com/paper/2409.08477/) (2024)
- [ADAM-SINDy: An Efficient Optimization Framework for Parameterized Nonlinear Dynamical System Identification](https://mlsanity.com/paper/2410.16528/) (2024)
- [A Neural Symbolic Model for Space Physics](https://mlsanity.com/paper/2503.07994/) (2025)
- [Representation Meets Optimization: Training PINNs and PIKANs for Gray-Box Discovery in Systems Pharmacology](https://mlsanity.com/paper/2504.07379/) (2025)
- [AI-Lorenz: A physics-data-driven framework for black-box and gray-box identification of chaotic systems with symbolic regression](https://mlsanity.com/paper/2312.14237/) (2023)
- [Nonlinear Discrete-Time Observers with Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.12360/) (2024)
- [Physics-Informed Neural Networks and Extensions](https://mlsanity.com/paper/2408.16806/) (2024)
- [GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Laws](https://mlsanity.com/paper/2410.22193/) (2024)
- [Spectral bias in physics-informed and operator learning: Analysis and mitigation guidelines](https://mlsanity.com/paper/2602.19265/) (2026)
- [CMINNs: Compartment Model Informed Neural Networks -- Unlocking Drug Dynamics](https://mlsanity.com/paper/2409.12998/) (2024)
- [Adapting Physics-Informed Neural Networks to Improve ODE Optimization in Mosquito Population Dynamics](https://mlsanity.com/paper/2406.05108/) (2024)
- [Physics-Informed Machine Learning in Biomedical Science and Engineering](https://mlsanity.com/paper/2510.05433/) (2025)
- [Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiology](https://mlsanity.com/paper/2408.07201/) (2024)
- [Are Two Hidden Layers Still Enough for the Physics-Informed Neural Networks?](https://mlsanity.com/paper/2412.19235/) (2024)
- [DAE-HardNet: A Physics Constrained Neural Network Enforcing Differential-Algebraic Hard Constraints](https://mlsanity.com/paper/2512.05881/) (2025)
- [A physics-informed neural network method for the approximation of slow invariant manifolds for the general class of stiff systems of ODEs](https://mlsanity.com/paper/2403.11591/) (2024)
- [FMEnets: Flow, Material, and Energy networks for non-ideal plug flow reactor design](https://mlsanity.com/paper/2505.20300/) (2025)
- [Next Generation Equation-Free Multiscale Modelling of Crowd Dynamics via Machine Learning](https://mlsanity.com/paper/2508.03926/) (2025)
- [Learning Coupled System Dynamics under Incomplete Physical Constraints and Missing Data](https://mlsanity.com/paper/2512.23761/) (2025)
- [Physics Informed Constrained Learning of Dynamics from Static Data](https://mlsanity.com/paper/2504.12675/) (2025)
- [Learning dynamical systems with biochemically informed neural ordinary differential equations](https://mlsanity.com/paper/2605.24170/) (2026)
- [Quantum-classical physics-informed Kolmogorov-Arnold networks for PDEs](https://mlsanity.com/paper/2606.20326/) (2026)
- [About rectified sigmoid function for enhancing the accuracy of Physics-Informed Neural Networks](https://mlsanity.com/paper/2412.20851/) (2024)
- [CBINNS: Cancer Biology-Informed Neural Network for Unknown Parameter Estimation and Missing Physics Identification](https://mlsanity.com/paper/2510.17920/) (2025)
- [Stability and Bifurcation Analysis of Nonlinear PDEs via Random Projection-based PINNs: A Krylov-Arnoldi Approach](https://mlsanity.com/paper/2603.21568/) (2026)
- [Meta-Inverse Physics-Informed Neural Networks for High-Dimensional Ordinary Differential Equations](https://mlsanity.com/paper/2605.03511/) (2026)
- [Physics-Informed Neural Networks for Chemotherapy Pharmacokinetics: Benchmarking the Clinical Estimator and Exposing Parameter Identifiability](https://mlsanity.com/paper/2606.12658/) (2026)
- [Quantum-Classical Physics-Informed Kolmogorov-Arnold Networks for Solving Fuzzy Differential Equations](https://mlsanity.com/paper/2608.08782/) (2026)

## Abstract (arXiv)

Discovering mathematical equations that govern physical and biological systems from observed data is a fundamental challenge in scientific research. We present a new physics-informed framework for parameter estimation and missing physics identification (gray-box) in the field of Systems Biology. The proposed framework -- named AI-Aristotle -- combines eXtreme Theory of Functional Connections (X-TFC) domain-decomposition and Physics-Informed Neural Networks (PINNs) with symbolic regression (SR) techniques for parameter discovery and gray-box identification. We test the accuracy, speed, flexibility and robustness of AI-Aristotle based on two benchmark problems in Systems Biology: a pharmacokinetics drug absorption model, and an ultradian endocrine model for glucose-insulin interactions. We compare the two machine learning methods (X-TFC and PINNs), and moreover, we employ two different symbolic regression techniques to cross-verify our results. While the current work focuses on the performance of AI-Aristotle based on synthetic data, it can equally handle noisy experimental data and can even be used for black-box identification in just a few minutes on a laptop. More broadly, our work provides insights into the accuracy, cost, scalability, and robustness of integrating neural networks with symbolic regressors, offering a comprehensive guide for researchers tackling gray-box identification challenges in complex dynamical systems in biomedicine and beyond.