# A Framework Based on Symbolic Regression Coupled with eXtended Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion from Data

arXiv: https://arxiv.org/abs/2305.10706 · HTML: https://mlsanity.com/paper/2305.10706/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 80 references that are in the map.
- First posted on arXiv 18 May 2023.

## Builds on (most cited first, 24 of 24)

- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [Learning two-phase microstructure evolution using neural operators and autoencoder architectures](https://mlsanity.com/paper/2204.07230/) (2022)
- [Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks](https://mlsanity.com/paper/2005.03596/) (2020)
- [Machine learning for metal additive manufacturing: Predicting temperature and melt pool fluid dynamics using physics-informed neural networks](https://mlsanity.com/paper/2008.13547/) (2020)
- [Coarse-scale PDEs from fine-scale observations via machine learning](https://mlsanity.com/paper/1909.05707/) (2019)
- [Phase-Field DeepONet: Physics-informed deep operator neural network for fast simulations of pattern formation governed by gradient flows of free-energy functionals](https://mlsanity.com/paper/2302.13368/) (2023)
- [SINDy-BVP: Sparse Identification of Nonlinear Dynamics for Boundary Value Problems](https://mlsanity.com/paper/2005.10756/) (2020)
- [Symbolic Regression by Exhaustive Search: Reducing the Search Space Using Syntactical Constraints and Efficient Semantic Structure Deduplication](https://mlsanity.com/paper/2109.13895/) (2021)
- [SINDy with Control: A Tutorial](https://mlsanity.com/paper/2108.13404/) (2021)
- [Physics-Informed Neural Networks for Shell Structures](https://mlsanity.com/paper/2207.14291/) (2022)
- [Revisiting PINNs: Generative Adversarial Physics-informed Neural Networks and Point-weighting Method](https://mlsanity.com/paper/2205.08754/) (2022)
- [Gray-box Adversarial Testing for Control Systems with Machine Learning Component](https://mlsanity.com/paper/1812.11958/) (2018)
- [Black and Gray Box Learning of Amplitude Equations: Application to Phase Field Systems](https://mlsanity.com/paper/2207.03954/) (2022)

## Built on by (most cited first, 18 of 18)

- [Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2501.16371/) (2025)
- [Integrating Neural Operators with Diffusion Models Improves Spectral Representation in Turbulence Modeling](https://mlsanity.com/paper/2409.08477/) (2024)
- [Correcting model misspecification in physics-informed neural networks (PINNs)](https://mlsanity.com/paper/2310.10776/) (2023)
- [AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identification](https://mlsanity.com/paper/2310.01433/) (2023)
- [Rethinking materials simulations: Blending direct numerical simulations with neural operators](https://mlsanity.com/paper/2312.05410/) (2023)
- [Predicting Crack Nucleation and Propagation in Brittle Materials Using Deep Operator Networks with Diverse Trunk Architectures](https://mlsanity.com/paper/2501.00016/) (2024)
- [Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networks](https://mlsanity.com/paper/2604.05230/) (2026)
- [Representation Meets Optimization: Training PINNs and PIKANs for Gray-Box Discovery in Systems Pharmacology](https://mlsanity.com/paper/2504.07379/) (2025)
- [NeuroSEM: A hybrid framework for simulating multiphysics problems by coupling PINNs and spectral elements](https://mlsanity.com/paper/2407.21217/) (2024)
- [Characterization of partial wetting by CMAS droplets using multiphase many-body dissipative particle dynamics and data-driven discovery based on PINNs](https://mlsanity.com/paper/2307.09142/) (2023)
- [No One-Size-Fits-All Neurons: Task-based Neurons for Artificial Neural Networks](https://mlsanity.com/paper/2405.02369/) (2024)
- [Meta-learning Loss Functions of Parametric Partial Differential Equations Using Physics-Informed Neural Networks](https://mlsanity.com/paper/2412.00225/) (2024)
- [Discovering equations from data: symbolic regression in dynamical systems](https://mlsanity.com/paper/2508.20257/) (2025)
- [Symmetry group based domain decomposition to enhance physics-informed neural networks for solving partial differential equations](https://mlsanity.com/paper/2404.18538/) (2024)
- [Crack Path Prediction with Operator Learning using Discrete Particle System data Generation](https://mlsanity.com/paper/2506.01976/) (2025)
- [NeuronSeek: On Stability and Expressivity of Task-driven Neurons](https://mlsanity.com/paper/2506.15715/) (2025)
- [GIMLET: Generalizable and Interpretable Model Learning through Embedded Thermodynamics](https://mlsanity.com/paper/2512.19936/) (2025)
- [OrthoReg: Orthogonal Regularization for Hybrid Symbolic-Neural Dynamical Systems](https://mlsanity.com/paper/2606.19145/) (2026)

## Abstract (arXiv)

We propose a framework and an algorithm to uncover the unknown parts of nonlinear equations directly from data. The framework is based on eXtended Physics-Informed Neural Networks (X-PINNs), domain decomposition in space-time, but we augment the original X-PINN method by imposing flux continuity across the domain interfaces. The well-known Allen-Cahn equation is used to demonstrate the approach. The Frobenius matrix norm is used to evaluate the accuracy of the X-PINN predictions and the results show excellent performance. In addition, symbolic regression is employed to determine the closed form of the unknown part of the equation from the data, and the results confirm the accuracy of the X-PINNs based approach. To test the framework in a situation resembling real-world data, random noise is added to the datasets to mimic scenarios such as the presence of thermal noise or instrument errors. The results show that the framework is stable against significant amount of noise. As the final part, we determine the minimal amount of data required for training the neural network. The framework is able to predict the correct form and coefficients of the underlying dynamical equation when at least 50% data is used for training.