mlsanityData to 14 Oct 2026

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

arXiv 2305.10706 ↗First posted 18 May 2023cond-mat.dis-nn cs.LGcited by 18

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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.

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

24 of 80 references.

References · 24

  1. 2018On the Spectral Bias of Neural Networkscited by 1,341
  2. 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
  3. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
  4. 2019AI Feynman: a Physics-Inspired Method for Symbolic Regressioncited by 548
  5. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 335
  6. 2017PDE-Net: Learning PDEs from Datacited by 311
  7. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 210
  8. 2017Sparse identification of nonlinear dynamics for model predictive control in the low-data limitcited by 140
  9. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 129
  10. 2020Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networkscited by 123
  11. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  12. 2021Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problemscited by 102
12 more
  1. 2022Learning two-phase microstructure evolution using neural operators and autoencoder architecturescited by 72
  2. 2020Physics-informed neural network for ultrasound nondestructive quantification of surface breaking crackscited by 68
  3. 2020Machine learning for metal additive manufacturing: Predicting temperature and melt pool fluid dynamics using physics-informed neural networkscited by 42
  4. 2019Coarse-scale PDEs from fine-scale observations via machine learningcited by 35
  5. 2023Phase-Field DeepONet: Physics-informed deep operator neural network for fast simulations of pattern formation governed by gradient flows of free-energy functionalscited by 26
  6. 2020SINDy-BVP: Sparse Identification of Nonlinear Dynamics for Boundary Value Problemscited by 22
  7. 2021Symbolic Regression by Exhaustive Search: Reducing the Search Space Using Syntactical Constraints and Efficient Semantic Structure Deduplicationcited by 17
  8. 2021SINDy with Control: A Tutorialcited by 15
  9. 2022Physics-Informed Neural Networks for Shell Structurescited by 13
  10. 2022Revisiting PINNs: Generative Adversarial Physics-informed Neural Networks and Point-weighting Methodcited by 10
  11. 2018Gray-box Adversarial Testing for Control Systems with Machine Learning Componentcited by 8
  12. 2022Black and Gray Box Learning of Amplitude Equations: Application to Phase Field Systemscited by 5

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Citing papers per year: 2023: 4; 2024: 6; 2025: 6; 2026: 2.

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  1. 2025Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networkscited by 42
  2. 2024Integrating Neural Operators with Diffusion Models Improves Spectral Representation in Turbulence Modelingcited by 36
  3. 2023Correcting model misspecification in physics-informed neural networks (PINNs)cited by 33
  4. 2023AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identificationcited by 29
  5. 2023Rethinking materials simulations: Blending direct numerical simulations with neural operatorscited by 25
  6. 2024Predicting Crack Nucleation and Propagation in Brittle Materials Using Deep Operator Networks with Diverse Trunk Architecturescited by 12
  7. 2026Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networkscited by 11
  8. 2025Representation Meets Optimization: Training PINNs and PIKANs for Gray-Box Discovery in Systems Pharmacologycited by 10
  9. 2024NeuroSEM: A hybrid framework for simulating multiphysics problems by coupling PINNs and spectral elementscited by 6
  10. 2023Characterization of partial wetting by CMAS droplets using multiphase many-body dissipative particle dynamics and data-driven discovery based on PINNscited by 4
  11. 2024No One-Size-Fits-All Neurons: Task-based Neurons for Artificial Neural Networkscited by 3
  12. 2024Meta-learning Loss Functions of Parametric Partial Differential Equations Using Physics-Informed Neural Networkscited by 2
6 more
  1. 2025Discovering equations from data: symbolic regression in dynamical systemscited by 2
  2. 2024Symmetry group based domain decomposition to enhance physics-informed neural networks for solving partial differential equationscited by 1
  3. 2025Crack Path Prediction with Operator Learning using Discrete Particle System data Generationcited by 0
  4. 2025NeuronSeek: On Stability and Expressivity of Task-driven Neuronscited by 0
  5. 2025GIMLET: Generalizable and Interpretable Model Learning through Embedded Thermodynamicscited by 0
  6. 2026OrthoReg: Orthogonal Regularization for Hybrid Symbolic-Neural Dynamical Systemscited by 0