---
title: "Symbolic Regression"
created: 2026-04-16
updated: 2026-04-17
type: concept
tags: [optimization, training, model]
sources: [raw/papers/odrzywolek-eml-universal-operator-2026.md]
---

# Symbolic Regression

**Symbolic regression** is a machine learning technique that discovers explicit mathematical expressions from data, rather than fitting fixed-form models. Unlike traditional regression (which optimizes parameters within a predetermined functional form), symbolic regression searches the space of possible equation structures.

## Core Problem

Given data points (xᵢ, yᵢ), find a closed-form expression f such that y ≈ f(x), where f is composed of elementary operations and functions.

**Key Distinction:**
- Traditional regression: y = β₀ + β₁x + β₂x² (form fixed, optimize β)
- Symbolic regression: Discover that y = sin(2πx) · e^(-x²) from data

## Traditional Approaches

### Genetic Programming

The dominant approach historically:
- **Representation**: Expression trees with heterogeneous nodes (+, -, ×, ÷, sin, exp, etc.)
- **Search**: Evolutionary algorithms (mutations, crossovers)
- **Fitness**: Mean squared error or complexity-penalized metrics
- **Tools**: Eureqa, gplearn, PySR

**Limitations:**
- Discrete search space (combinatorial explosion)
- Slow convergence for complex expressions
- No gradient information
- Brittle to hyperparameters

### Sparse Regression (SINDy)

- Assumes sparse linear combination from a library of candidate functions
- Uses LASSO/sparse optimization
- Faster but limited to linear combinations of basis functions

## Gradient-Based Approaches

Recent work enables differentiable symbolic regression:

### EML Trees (2026)

[[eml-operator|Odrzywołek's EML representation]] enables gradient-based optimization:
- Uniform tree structure (all nodes are `eml` operators)
- Fully differentiable
- Optimizable with standard deep learning optimizers (Adam)
- Can recover exact closed forms at shallow depths (≤4)

### Neural Symbolic Methods

- **AI Feynman**: Combines neural network fitting with symbolic property testing
- **Symbolic GPT**: Transformer-based generation of expressions
- **Deep Symbolic Regression**: Neural networks predicting expression trees

## Evaluation Metrics

1. **Accuracy**: R², MSE, NMSE on held-out data
2. **Complexity**: Number of nodes, operators, or description length
3. **Pareto Frontier**: Trade-off between accuracy and simplicity
4. **Exact Recovery**: Whether the true underlying formula is found
5. **Generalization**: Performance on out-of-distribution data

## Applications

| Domain | Example |
|--------|---------|
| Physics | Discovering force laws, equations of state |
| Chemistry | Reaction kinetics, structure-property relationships |
| Biology | Population dynamics, gene regulatory networks |
| Engineering | System identification, control laws |
| Finance | Discovering pricing formulas, risk models |

## Challenges

1. **Scalability**: Exponential growth of expression space with size
2. **Noise Sensitivity**: Overfitting to data noise
3. **Non-uniqueness**: Multiple expressions may fit data equally well
4. **Dimensional Analysis**: Incorporating physical units/constraints
5. **Interpretability**: Balancing accuracy with human-understandable forms

## Future Directions

- Integration with large language models for prior knowledge
- Physics-informed constraints (conservation laws, symmetries)
- Multi-objective optimization (accuracy, simplicity, generalization)
- Real-time/online symbolic regression
- Human-in-the-loop discovery workflows

## Related Concepts

- [[eml-operator]]: A universal operator enabling gradient-based symbolic regression
- [[andrzej-odrzywolek]]: Researcher who discovered the EML universal operator
- [[computerized-adaptive-testing]]: CAT 中的动态选题策略与符号回归中的自适应搜索在"探索-利用权衡"上有结构相似性
