# Closed-form Solutions: A New Perspective on Solving Differential Equations

arXiv: https://arxiv.org/abs/2405.14620 · HTML: https://mlsanity.com/paper/2405.14620/

## 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 (most cited first, 21 of 21)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [Learning Equations for Extrapolation and Control](https://mlsanity.com/paper/1806.07259/) (2018)
- [Neural Optimizer Search with Reinforcement Learning](https://mlsanity.com/paper/1709.07417/) (2017)
- [Transformer-based Planning for Symbolic Regression](https://mlsanity.com/paper/2303.06833/) (2023)
- [Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree search](https://mlsanity.com/paper/2205.13134/) (2022)
- [Encoding physics to learn reaction-diffusion processes](https://mlsanity.com/paper/2106.04781/) (2021)
- [Deep Learning and Symbolic Regression for Discovering Parametric Equations](https://mlsanity.com/paper/2207.00529/) (2022)
- [Finite Expression Method for Solving High-Dimensional Partial Differential Equations](https://mlsanity.com/paper/2206.10121/) (2022)
- [A Neural-Guided Dynamic Symbolic Network for Exploring Mathematical Expressions from Data](https://mlsanity.com/paper/2309.13705/) (2023)
- [Symbolic Regression via Control Variable Genetic Programming](https://mlsanity.com/paper/2306.08057/) (2023)
- [Symbolic Regression for PDEs using Pruned Differentiable Programs](https://mlsanity.com/paper/2303.07009/) (2023)
- [Genetic Programming Based Symbolic Regression for Analytical Solutions to Differential Equations](https://mlsanity.com/paper/2302.03175/) (2023)

## Built on by (most cited first, 2 of 2)

- [PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs](https://mlsanity.com/paper/2502.00963/) (2025)
- [SymPlex: A Structure-Aware Transformer for Symbolic PDE Solving](https://mlsanity.com/paper/2602.03816/) (2026)

## 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.