Closed-form Solutions: A New Perspective on Solving Differential Equations
Not on the map. Why?
Abstract (arXiv)
The quest for analytical solutions to differential equations has traditionally been constrained by the need for extensive mathematical expertise. Machine learning methods like genetic algorithms have shown promise in this domain, but are hindered by significant computational time and the complexity of their derived solutions. This paper introduces SSDE (Symbolic Solver for Differential Equations), a novel reinforcement learning-based approach that derives symbolic closed-form solutions for various differential equations. Evaluations across a diverse set of ordinary and partial differential equations demonstrate that SSDE outperforms existing machine learning methods, delivering superior accuracy and efficiency in obtaining analytical solutions.
Key facts. Cited by 2 papers in the map (data through 14 Oct 2026). Builds on 21 of its 55 references that are in the map. First posted on arXiv 23 May 2024.
Builds on
21 of 55 references.
References · 21
- 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
- 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
- 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 427
- 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 413
- 2019Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradientscited by 309
- 2022Transformer for Partial Differential Equations' Operator Learningcited by 283
- 2021Physics-informed neural networks with hard constraints for inverse designcited by 280
- 2021Contemporary Symbolic Regression Methods and their Relative Performancecited by 215
- 2022End-to-end symbolic regression with transformerscited by 203
- 2021Neural Symbolic Regression that Scalescited by 192
- 2018Learning Equations for Extrapolation and Controlcited by 156
- 2017Neural Optimizer Search with Reinforcement Learningcited by 82
9 more
- 2023Transformer-based Planning for Symbolic Regressioncited by 79
- 2022Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree searchcited by 66
- 2021Encoding physics to learn reaction-diffusion processescited by 61
- 2022Deep Learning and Symbolic Regression for Discovering Parametric Equationscited by 23
- 2022Finite Expression Method for Solving High-Dimensional Partial Differential Equationscited by 21
- 2023A Neural-Guided Dynamic Symbolic Network for Exploring Mathematical Expressions from Datacited by 16
- 2023Symbolic Regression via Control Variable Genetic Programmingcited by 9
- 2023Symbolic Regression for PDEs using Pruned Differentiable Programscited by 7
- 2023Genetic Programming Based Symbolic Regression for Analytical Solutions to Differential Equationscited by 6
Built on by
Citing papers per year: 2025: 1; 2026: 1.
2 papers.
Similar, not cited
Close in wording, no citation either way
- 2025Neuro-Symbolic AI for Analytical Solutions of Differential Equations
- 2025Symbolic Neural Ordinary Differential Equations
- 2025Deep Symbolic Optimization: Reinforcement Learning for Symbolic Mathematics
- 2024Symbolic Equation Solving via Reinforcement Learning
- 2026Reinforcement Learning for Symbolic Equation Solving
- 2026Enhancing Transformer Representations of Symbolic ODE Expressions
- 2024Estimating unknown parameters in differential equations with a reinforcement learning based PSO method
- 2024Identifying Ordinary Differential Equations for Data-efficient Model-based Reinforcement Learning