README.md

---
tags:
- synthetic-data
- physics
- symbolic-regression
- scientific-discovery
- physical-theories
license: mit
---

# SynPAT: Generating Synthetic Physical Theories with Data

This is the Hugging Face dataset entry for **SynPAT**, a synthetic theory and data generation system developed for the paper:  
**SynPAT: Generating Synthetic Physical Theories with Data**  
GitHub: [https://github.com/jlenchner/theorizer](https://github.com/jlenchner/theorizer)

SynPAT generates symbolic physical systems and corresponding synthetic data to benchmark symbolic regression and scientific discovery algorithms. Each synthetic system includes symbolic equations, dimensionally consistent variables/constants, algebraic consequences, and both clean and noisy datasets.

---

## Dataset Structure

The dataset is organized hierarchically by configuration and system index:

```
dataset/{config}/System_{n}/
```


Where:
- `{v}` ∈ {6, 7, 8, 9} is the number of symbolic variables
- `{d}` ∈ {2, 3, 4} is the number of derivatives (among the symbolic variables)
- `{e}` ∈ {4, 5, 6} is the number of equations in the system
- `{n}` ∈ {1, 2, 3} indexes the individual system instances per configuration

Each `System_{n}` directory contains exactly 18 files:

### Symbolic System and Consequence:
- `system.txt` — base symbolic system with metadata including variable types, units, constants, and equations
- `consequence.txt` — derived consequence equation (single polynomial) with associated metadata

### Noiseless Data:
- `system.dat` — clean numerical dataset corresponding to `system.txt`
- `consequence.dat` — clean data for the consequence polynomial

### Noisy Data:
For each noise level ε ∈ {0.001, 0.01, 0.05, 0.1}, the following files are included:
- `system_ε.dat` — noisy system data with Gaussian noise of standard deviation ε
- `consequence_ε.dat` — noisy consequence data with Gaussian noise of standard deviation ε

This results in:
- 4 `system_ε.dat` files
- 4 `consequence_ε.dat` files

### Replacement Systems:
- `replacement_1.txt` through `replacement_5.txt` — versions of `system.txt` where one axiom has been replaced, used for consequence prediction benchmarking under ablation.

### Summary of File Count per System:
- 2 base files: `system.txt`, `consequence.txt`
- 2 clean data files: `system.dat`, `consequence.dat`
- 8 noisy data files: `system_ε.dat`, `consequence_ε.dat` for 4 ε values
- 5 replacement files: `replacement_1.txt` through `replacement_5.txt`

**Total per system:** 2 + 2 + 8 + 5 = **17 files**

Each configuration folder (`vars_{v}_derivs_{d}_eqns_{e}`) contains exactly three systems: `System_1`, `System_2`, `System_3`.

The full dataset spans all 4 × 3 × 3 = **36 configurations**, totaling **108 systems** and **1836 files**.


## File Format Example

### system.txt 

`system.txt` contains the symbolic system, including variables, constants, derivatives, equations, and units:

```text
Variables: ['Fc', 'Fg', 'W', 'd1', 'd2', 'm1']  
Constants: ['G', 'c']  
Derivatives: ['dx1dt', 'd2x1dt2', 'dx2dt']  
Equations:  
dx1dt*dx2dt*d2*m1 - W*d1  
d2x1dt2*W*m1 - Fc*Fg*d1  
2*c*Fg + dx2dt*Fc - dx2dt*Fg  
G*d2x1dt2*d2*m1 + G*Fc*d2 - G*W + 4*d2x1dt2*dx1dt^2*d1^2  
-d2x1dt2*Fc*m1 + d2x1dt2*Fg*m1 + Fc*Fg  
Units of Measure of Variables: ['s^(-2)*kg*m', 's^(-2)*kg*m', 'm^2*s^(-2)*kg', 'm', 'm', 'kg']  
Units of Measure of Constants: ['1/kg*m^3*s^(-2)', '1/s*m']  
Units of Measure of Derivatives: ['1/s*m', 's^(-2)*m', '1/s*m']  
Units of Measure of Equations:  
m^3*s^(-2)*kg  
kg^2*m^3*s^(-4)  
m^2*s^(-3)*kg  
m^5*s^(-4)  
kg^2*m^2*s^(-4)
```

### consequence.txt 

`consequence.txt` includes **additional metadata** for data generation and symbolic regression tasks:

```text
Variables: ['Fc', 'Fg', 'W', 'd1', 'd2', 'm1']  
Constants: ['G', 'c']  
Derivatives: ['dx1dt', 'd2x1dt2', 'dx2dt']  
Equations:  
dx1dt*dx2dt*d2*m1 - W*d1  
d2x1dt2*W*m1 - Fc*Fg*d1  
2*c*Fg + dx2dt*Fc - dx2dt*Fg  
G*d2x1dt2*d2*m1 + G*Fc*d2 - G*W + 4*d2x1dt2*dx1dt^2*d1^2  
-d2x1dt2*Fc*m1 + d2x1dt2*Fg*m1 + Fc*Fg  
Units of Measure of Variables: ['s^(-2)*kg*m', 's^(-2)*kg*m', 'm^2*s^(-2)*kg', 'm', 'm', 'kg']  
Units of Measure of Constants: ['1/kg*m^3*s^(-2)', '1/s*m']  
Units of Measure of Derivatives: ['1/s*m', 's^(-2)*m', '1/s*m']  
Units of Measure of Equations:  
m^3*s^(-2)*kg  
kg^2*m^3*s^(-4)  
m^2*s^(-3)*kg  
m^5*s^(-4)  
kg^2*m^2*s^(-4)

Measured Variables: ['d1', 'Fg', 'm1', 'W']  
Observed Constants: []  
Measured Derivatives: ['d2x1dt2']  

Target Polynomial:  
d1*m1*d2x1dt2*Fg^2 - m1^2*d2x1dt2^2*W + m1*d2x1dt2*W*Fg
```

### replacement_i.txt

`replacement_i.txt` has the same format as `system.txt` but with one axiom replaced by a different, dimensionally consistent axiom
```text
Variables: ['Fc', 'Fg', 'W', 'd1', 'd2', 'm1']
Constants: ['G', 'c']
Derivatives: ['dx1dt', 'd2x1dt2', 'dx2dt']
Equations:
dx1dt*dx2dt*d2*m1 - W*d1
>>> c*Fc*d2 + c*Fg*d1 - dx1dt*Fg*d1 + dx2dt*Fg*d2 + dx2dt*W   <<< 
2*c*Fg + dx2dt*Fc - dx2dt*Fg
G*d2x1dt2*d2*m1 + G*Fc*d2 - G*W + 4*d2x1dt2*dx1dt^2*d1^2
-d2x1dt2*Fc*m1 + d2x1dt2*Fg*m1 + Fc*Fg
Units of Measure of Variables: ['s^(-2)*kg*m', 's^(-2)*kg*m', 'm^2*s^(-2)*kg', 'm', 'm', 'kg']
Units of Measure of Constants: ['1/kg*m^3*s^(-2)', '1/s*m']
Units of Measure of Derivatives: ['1/s*m', 's^(-2)*m', '1/s*m']
Units of Measure of Equations:
m^3*s^(-2)*kg
m^3*s^(-3)*kg
m^2*s^(-3)*kg
m^5*s^(-4)
kg^2*m^2*s^(-4)
```

### Data Files

Each `.dat` file contains rows of numerical data corresponding to samples used for learning or evaluation. Each row represents one data point.

- For **`consequence.dat`** and its noisy versions (e.g., `consequence_0.1.dat`), **the columns correspond to the observed constants, followed by measured derivatives, and then measured variables**, in the exact order as specified in `consequence.txt`.

Example row from `consequence_0.1.dat`:

```text
1.925664644193098241e+00 2.872700594236812677e+00 ...
```

- For **`system.dat`** and its noisy versions (e.g., `system_0.1.dat`), **the first line is a header specifying the variable ordering**, followed by rows of numerical data where each column matches the respective variable in the header.

Example header and data snippet from `system_0.1.dat`:

```text
G c dx1dt d2x1dt2 dx2dt Fc W d1 Fg m1 d2
1.000000 1.0000000 ......
```

---

## Authors

- **Karan Srivastava** — University of Wisconsin-Madison (ksrivastava4@wisc.edu)  
- **Jonathan Lenchner** — IBM T.J. Watson Research Center (lenchner@us.ibm.com)  
- **Joao Goncalves** — IBM T.J. Watson Research Center (jpgoncal@us.ibm.com)  
- **Lior Horesh** — IBM T.J. Watson Research Center (lhoresh@us.ibm.com)  

---

## License

MIT License

---

For issues, questions, or contributions, please visit:  
[https://github.com/jlenchner/theorizer](https://github.com/jlenchner/theorizer)