# SNIP++: Fine-Grained Symbolic-Numerical Alignment for Symbolic Regression

arXiv: https://arxiv.org/abs/2609.31965 · HTML: https://mlsanity.com/paper/2609.31965/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 33 references that are in the map.
- First posted on arXiv 25 Sep 2026.

## Builds on (most cited first, 15 of 15)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [Does CLIP Bind Concepts? Probing Compositionality in Large Image Models](https://mlsanity.com/paper/2212.10537/) (2022)
- [FG-CLIP: Fine-Grained Visual and Textual Alignment](https://mlsanity.com/paper/2505.05071/) (2025)
- [Transformer-based Planning for Symbolic Regression](https://mlsanity.com/paper/2303.06833/) (2023)
- [SymFormer: End-to-end symbolic regression using transformer-based architecture](https://mlsanity.com/paper/2205.15764/) (2022)
- [Equivariant Similarity for Vision-Language Foundation Models](https://mlsanity.com/paper/2303.14465/) (2023)
- [SNIP: Bridging Mathematical Symbolic and Numeric Realms with Unified Pre-training](https://mlsanity.com/paper/2310.02227/) (2023)
- [Learning Continuous Semantic Representations of Symbolic Expressions](https://mlsanity.com/paper/1611.01423/) (2016)
- [Symbolic regression via MDLformer-guided search: from minimizing prediction error to minimizing description length](https://mlsanity.com/paper/2411.03753/) (2024)
- [MMSR: Symbolic Regression is a Multi-Modal Information Fusion Task](https://mlsanity.com/paper/2402.18603/) (2024)
- [Analyzing Generalization in Pre-Trained Symbolic Regression](https://mlsanity.com/paper/2509.19849/) (2025)
- [Multi-Modal Learning meets Genetic Programming: Analyzing Alignment in Latent Space Optimization](https://mlsanity.com/paper/2604.08324/) (2026)

## Abstract (arXiv)

Mathematical expressions and the numerical behavior they produce are two views of the same underlying function, and connecting them is central to scientific discovery. Symbolic Regression (SR) relies on this connection directly: it searches for an expression that reproduces a given behavior. Recent multi-modal models learn this connection by embedding symbolic expressions and their numerical behavior in a shared representation space. We show that this embedding space is only globally aligned: complete expressions correspond to complete behaviors, but the contribution of individual parts of an expression is not represented. This granularity gap leaves the model unable to tell how a local edit to an expression changes its behavior, the central operation in SR. We introduce a compositional alignment method that closes this gap: a structural positional encoding exposes the substructure of an expression to the encoder, and a multi-granularity contrastive objective grounds each subexpression in the behavior it produces before propagating this grounding to the full expression. The resulting representations close much of the modality gap between symbolic and numerical embeddings, reliably distinguish the effects of local edits that the original alignment cannot, and transfer to external SR corpora.