# Neuro-Symbolic AI for Analytical Solutions of Differential Equations

arXiv: https://arxiv.org/abs/2502.01476 · HTML: https://mlsanity.com/paper/2502.01476/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 44 references that are in the map.
- First posted on arXiv 3 Feb 2025.

## Builds on (most cited first, 16 of 16)

- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Grammar Variational Autoencoder](https://mlsanity.com/paper/1703.01925/) (2017)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [Symbolic Regression is NP-hard](https://mlsanity.com/paper/2207.01018/) (2022)
- [SymFormer: End-to-end symbolic regression using transformer-based architecture](https://mlsanity.com/paper/2205.15764/) (2022)
- [Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolation](https://mlsanity.com/paper/2404.12355/) (2024)
- [Generative AI for fast and accurate statistical computation of fluids](https://mlsanity.com/paper/2409.18359/) (2024)
- [Universal Physics Transformers: A Framework For Efficiently Scaling Neural Operators](https://mlsanity.com/paper/2402.12365/) (2024)
- [UPS: Efficiently Building Foundation Models for PDE Solving via Cross-Modal Adaptation](https://mlsanity.com/paper/2403.07187/) (2024)
- [Incorporating domain knowledge into neural-guided search](https://mlsanity.com/paper/2107.09182/) (2021)

## Built on by (most cited first, 1 of 1)

- [Grammar-based Ordinary Differential Equation Discovery](https://mlsanity.com/paper/2504.02630/) (2025)

## Abstract (arXiv)

Analytical solutions to differential equations offer exact, interpretable insight but are rarely available because discovering them requires expert intuition or exhaustive search of combinatorial spaces. We introduce SIGS, a neuro-symbolic framework for equation-driven closed-form solution discovery. SIGS uses a context-free grammar to generate mathematically valid and physically meaningful building blocks, with a user-specified Ansatz prescribing how these blocks combine, embeds them into a topology-regularised continuous latent manifold, and searches this manifold in two stages: structure selection followed by coefficient refinement using gradient descent, scoring candidates only against the PDE residual and prescribed boundary and initial conditions. This design unifies symbolic reasoning with numerical optimization; the grammar constrains candidate solution blocks to be proper by construction, while the latent search makes exploration tractable and data-free. SIGS is the first neuro-symbolic method to (i) recover analytical solutions for coupled nonlinear PDE systems, (ii) discover equivalent symbolic forms when the grammar lacks the natural primitives, and (iii) produce accurate symbolic approximations for PDEs lacking known closed-form solutions. Overall, SIGS improves over existing symbolic methods by orders of magnitude in both accuracy and runtime across standard PDE benchmarks.