mlsanityData to 7 Oct 2026

Deep Learning and Symbolic Regression for Discovering Parametric Equations

arXiv 2207.00529 ↗First posted 1 Jul 2022cs.LG cs.SC physics.comp-phcited by 22

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Abstract (arXiv)

Symbolic regression is a machine learning technique that can learn the governing formulas of data and thus has the potential to transform scientific discovery. However, symbolic regression is still limited in the complexity and dimensionality of the systems that it can analyze. Deep learning on the other hand has transformed machine learning in its ability to analyze extremely complex and high-dimensional datasets. We propose a neural network architecture to extend symbolic regression to parametric systems where some coefficient may vary but the structure of the underlying governing equation remains constant. We demonstrate our method on various analytic expressions, ODEs, and PDEs with varying coefficients and show that it extrapolates well outside of the training domain. The neural network-based architecture can also integrate with other deep learning architectures so that it can analyze high-dimensional data while being trained end-to-end. To this end we integrate our architecture with convolutional neural networks to analyze 1D images of varying spring systems.

Key facts. Cited by 22 papers in the map (data through 7 Oct 2026). Builds on 32 of its 50 references that are in the map. First posted on arXiv 1 Jul 2022.

Builds on

32 of 50 references.

References · 32

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20 more
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Citing papers per year: 2023: 2; 2024: 5; 2025: 9; 2026: 6.

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  1. 2024Training Stiff Neural Ordinary Differential Equations with Implicit Single-Step Methodscited by 11
  2. 2025BEDI: A Comprehensive Benchmark for Evaluating Embodied Agents on UAVscited by 10
  3. 2023Symbolic Regression for PDEs using Pruned Differentiable Programscited by 6
  4. 2023Bayesian polynomial neural networks and polynomial neural ordinary differential equationscited by 6
  5. 2024Training Stiff Neural Ordinary Differential Equations with Explicit Exponential Integration Methodscited by 5
  6. 2025Bridging the Domain Gap in Equation Distillation with Reinforcement Feedbackcited by 4
  7. 2024No One-Size-Fits-All Neurons: Task-based Neurons for Artificial Neural Networkscited by 3
  8. 2026Data-driven discovery of governing differential equations across physical systemscited by 3
  9. 2024Closed-form Solutions: A New Perspective on Solving Differential Equationscited by 2
  10. 2025A Neural Operator-Based Approach to Symbolic Discovery of PDEscited by 2
  11. 2025Automated Discovery of Conservation Laws via Hybrid Neural ODE-Transformerscited by 2
  12. 2024A Personalised Learning Tool for Physics Undergraduate Students Built On a Large Language Model for Symbolic Regressioncited by 1
10 more
  1. 2025Interpretable Non-linear Survival Analysis with Evolutionary Symbolic Regressioncited by 1
  2. 2025CTSR: Cartesian tensor-based sparse regression for data-driven discovery of high-dimensional invariant governing equationscited by 1
  3. 2025Data-Efficient Symbolic Regression via Foundation Model Distillationcited by 1
  4. 2025Optimized Architectures for Kolmogorov-Arnold Networkscited by 1
  5. 2026SymPlex: A Structure-Aware Transformer for Symbolic PDE Solvingcited by 1
  6. 2026Learning Physical Operators using Neural Operatorscited by 1
  7. 2025Sparse Interpretable Deep Learning with LIES Networks for Symbolic Regressioncited by 0
  8. 2026Deep Variational Inference Symbolic Regressioncited by 0
  9. 2026Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fieldscited by 0
  10. 2026SMILE: Bridging Continuous Optimization and Discrete Symbolic Recoverycited by 0