--- 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)