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Official PyTorch implementation of PSE/PSRN: Fast and efficient symbolic expression discovery through parallelized symbolic enumeration. Evaluates millions of expressions simultaneously on GPU with automated subtree reuse.

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# Discovering physical laws with parallel symbolic enumeration

*Official implementation of PSE with its core PSRN (Parallel Symbolic Regression Network) module*

**Authors:** Kai Ruan, Yilong Xu, Ze-Feng Gao, Yike Guo, Hao Sun, Ji-Rong Wen, Yang Liu

[![Article](https://img.shields.io/badge/Article-Nature_Computational_Science-E30613.svg?style=flat-square&logo=nature)](https://www.nature.com/articles/s43588-025-00904-8)
[![DeepWiki](https://img.shields.io/badge/DeepWiki-x66ccff%2FPSRN-silver.svg?logo=data:image/png;base64,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)](https://deepwiki.com/x66ccff/PSRN)
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![License](https://img.shields.io/badge/License-MIT-2196F3.svg)
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<!-- ![fig1.png](https://raw.githubusercontent.com/intell-sci-comput/PSE/main/assets/fig1.png)

![SRbench.png](https://raw.githubusercontent.com/intell-sci-comput/PSE/main/assets/SRbench.png) -->

![PSRN.jpg](https://raw.githubusercontent.com/intell-sci-comput/PSE/main/assets/PSRN.jpg)

This repository contains the official PyTorch implementation of PSE (Parallel Symbolic Enumeration): A fast and efficient symbolic expression discovery method powered by PSRN (Parallel Symbolic Regression Network). PSRN evaluates millions of symbolic expressions simultaneously on GPU with automated subtree reuse.

## News

[2026/01/30] 🎉🎉🎉 Our paper has been selected as the cover article for Nature Computational Science, Volume 6, Issue 1, January 2026!
<div align="center">
  <img src="./assets/ncs_cover.png" alt="Nature Computational Science Cover" style="max-width: 80%; height: auto;">
</div>

## 📜 Docs

[![DeepWiki](https://img.shields.io/badge/DeepWiki-x66ccff%2FPSRN-silver.svg?logo=data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACwAAAAyCAYAAAAnWDnqAAAAAXNSR0IArs4c6QAAA05JREFUaEPtmUtyEzEQhtWTQyQLHNak2AB7ZnyXZMEjXMGeK/AIi+QuHrMnbChYY7MIh8g01fJoopFb0uhhEqqcbWTp06/uv1saEDv4O3n3dV60RfP947Mm9/SQc0ICFQgzfc4CYZoTPAswgSJCCUJUnAAoRHOAUOcATwbmVLWdGoH//PB8mnKqScAhsD0kYP3j/Yt5LPQe2KvcXmGvRHcDnpxfL2zOYJ1mFwrryWTz0advv1Ut4CJgf5uhDuDj5eUcAUoahrdY/56ebRWeraTjMt/00Sh3UDtjgHtQNHwcRGOC98BJEAEymycmYcWwOprTgcB6VZ5JK5TAJ+fXGLBm3FDAmn6oPPjR4rKCAoJCal2eAiQp2x0vxTPB3ALO2CRkwmDy5WohzBDwSEFKRwPbknEggCPB/imwrycgxX2NzoMCHhPkDwqYMr9tRcP5qNrMZHkVnOjRMWwLCcr8ohBVb1OMjxLwGCvjTikrsBOiA6fNyCrm8V1rP93iVPpwaE+gO0SsWmPiXB+jikdf6SizrT5qKasx5j8ABbHpFTx+vFXp9EnYQmLx02h1QTTrl6eDqxLnGjporxl3NL3agEvXdT0WmEost648sQOYAeJS9Q7bfUVoMGnjo4AZdUMQku50McDcMWcBPvr0SzbTAFDfvJqwLzgxwATnCgnp4wDl6Aa+Ax283gghmj+vj7feE2KBBRMW3FzOpLOADl0Isb5587h/U4gGvkt5v60Z1VLG8BhYjbzRwyQZemwAd6cCR5/XFWLYZRIMpX39AR0tjaGGiGzLVyhse5C9RKC6ai42ppWPKiBagOvaYk8lO7DajerabOZP46Lby5wKjw1HCRx7p9sVMOWGzb/vA1hwiWc6jm3MvQDTogQkiqIhJV0nBQBTU+3okKCFDy9WwferkHjtxib7t3xIUQtHxnIwtx4mpg26/HfwVNVDb4oI9RHmx5WGelRVlrtiw43zboCLaxv46AZeB3IlTkwouebTr1y2NjSpHz68WNFjHvupy3q8TFn3Hos2IAk4Ju5dCo8B3wP7VPr/FGaKiG+T+v+TQqIrOqMTL1VdWV1DdmcbO8KXBz6esmYWYKPwDL5b5FA1a0hwapHiom0r/cKaoqr+27/XcrS5UwSMbQAAAABJRU5ErkJggg==)](https://deepwiki.com/x66ccff/PSRN)

https://deepwiki.com/x66ccff/PSRN

## 📥 Installation

**Prerequisite**: Python >=3.9, <=3.12

```bash
pip install psrn
```

Or build from source (https://github.com/x66ccff/PSRN):

```bash
pip install git+https://github.com/x66ccff/PSRN
```

## 🚀 Quickstart with Custom Data

Now, you can use `psrn-run` with custom data, use the following arguments:

### Options
```bash
  -q, --csvpath TEXT           path to custom csv file

-l, --operators         operator library (e.g., "['Add','Mul','Identity','Tanh','Abs']")
  --n_symbol_layers            number of symbol layers (default=3)
  -i, --n_inputs INTEGER       PSRN input size (n variables + n constants)
  -c, --use_constant BOOLEAN   use const in PSE

--use_cpu BOOLEAN            use cpu
  -g, --gpu_index INTEGER      gpu index used

--time_limit INTEGER         time limit (s)
  
  -d, --n_down_sample INTEGER  n sample to downsample in PSRN for speeding up
  -s, --seed INTEGER           seed
  -k, --topk INTEGER           number of best expressions to take from PSRN to
                               fit
  
  -o, --probe TEXT             expression probe, string, PSE will stop if
                               probe is in pf

--experiment_name TEXT       experiment_name
  --help                       Show this message and exit.
```

Examples
    ========
    >>> #   input data
    >>> #  ____^______
    >>> # /           |
    >>> # ['x',      'y',   'x+x',        '1.5',      '3']
    >>> # <-n_variables->
    >>> #                 <-n_cross->
    >>> #                            <-trying_const_num->
    >>> # <-----------------n_inputs--------------------->

For more detailed parameter settings, please use `psrn-run --help`

### 📝 CLI Examples

> [!TIP]
> - The last column of the csv should be the target variable
> - If using a version of PyTorch below 2.0, an error may occur during the `torch.topk` operation.
> - The experiments were performed on servers with Nvidia A100 (80GB) and Intel(R) Xeon(R) Platinum 8380 CPUs @ 2.30GHz.
> - We recommend using a high-memory GPU as smaller cards may encounter CUDA memory errors under our experimental settings. If you experience memory issues, consider reducing the number of input slots or opting for `semi_koza` operator sets (e.g., replacing `"Sub"` and `"Div"` with `"SemiSub"` and `"SemiDiv"`) or `basic` operator sets (e.g., replacing `"Sub"` and `"Div"` with `"Neg"` and `"Inv"`).

To run the script with build-in custom data with an expression probe (the algorithm will stop when it finds the expression or its symbolic equivalents):

```bash
psrn-run --csvpath ./your_data.csv -g 0 -i 5 -c False --probe "(exp(x0)-exp(-x0))/2"
```

Without an expression probe:

```bash
psrn-run --csvpath ./your_data.csv -g 0 -i 5 -c False
```

For limited VRAM (or when the ground truth expression is expected to be simple):

```bash
psrn-run --csvpath ./your_data.csv -g 0 -i 3 -c False --probe "(exp(x0)-exp(-x0))/2"
```

To customize the operator library:

```bash
psrn-run --csvpath ./your_data.csv -g 0 -i 5 -c False --probe "(exp(x0)-exp(-x0))/2" -l "['Add','Mul','Identity','Tanh','Abs']"
```

For custom data paths and operators:

```bash
psrn-run --csvpath ./your_data.csv -g 0 -i 5 -c False -l "['Add','Mul','SemiSub','SemiDiv','Identity']" 
```

**You can also reduce the number of layers to save VRAM (default is 3)
so that you can use more inputs  (e.g. 70 input PSRN, the rest 20 will be cross subtrees)**

```bash
psrn-run --csvpath ./50_cols_of_x_data.csv --n_symbol_layers 2 -g 0 -i 70 -c False -l "['Add','Mul','SemiSub','SemiDiv','Identity']" 
```

If you want to change the source code, please use the following steps:

clone the repo first 
```bash
git clone https://github.com/x66ccff/PSRN
```

then install the repo with edit mode
```bash
cd PSRN
pip install -e .
```

The `cli.py` is the entrance of the code, and you can change the `PSRN_Regressior` in `regressor.py`

## 📝 Python Examples

```python
import os
import click
import time
import numpy as np
import sympy as sp
import pandas as pd

default_csv = os.path.join(os.path.dirname(__file__), 'data', 'custom_data.csv')

@click.command()
@click.option("--experiment_name", default="_", type=str, help="experiment_name")
@click.option("--gpu_index", "-g", default=0, type=int, help="gpu index used")
@click.option("--operators","-l",default="['Add','Mul','Sub','Div','Identity','Sin','Cos','Exp','Log']",help="operator library")
@click.option("--n_symbol_layers",default=3,type=int,help="number of symbol layers (default=3)")
@click.option("--n_down_sample","-d",default=100,type=int,help="n sample to downsample in PSRN for speeding up")
@click.option("--n_inputs","-i",default=5,type=int,help="PSRN input size (n variables + n constants)")
@click.option("--seed", "-s", default=0, type=int, help="seed")
@click.option("--topk","-k",default=10,type=int,help="number of best expressions to take from PSRN to fit")
@click.option("--use_constant", "-c", default=False, type=bool, help="use const in PSE")
@click.option("--probe","-o",default=None,type=str,help="expression probe, string, PSE will stop if probe is in pf")
@click.option("--csvpath","-q",default=default_csv,type=str,help="path to custom csv file")
@click.option("--use_cpu",default=False,type=bool,help="use cpu")
@click.option("--time_limit", default=3600, type=int, help="time limit (s)")
def main(experiment_name, gpu_index, operators, n_down_sample, n_inputs, seed, topk, use_constant, probe, csvpath, use_cpu, time_limit):
    if not use_cpu:
        os.environ["CUDA_VISIBLE_DEVICES"] = str(gpu_index)

import torch
    from psrn import PSRN_Regressor

if not use_cpu:
        device = torch.device("cuda" if torch.cuda.is_available() else "cpu")
    else:
        device = torch.device("cpu")

operators = eval(operators)
    df = pd.read_csv(csvpath, header=None)

Input = df.values[:, :-1].reshape(len(df), -1)
    Output = df.values[:, -1].reshape(len(df), 1)

variables_name = [f"x{i}" for i in range(Input.shape[1])]

regressor = PSRN_Regressor(
        variables=variables_name,
        use_const=use_constant,
        n_symbol_layers=n_symbol_layers,
        device=device,
        token_generator_config={
            "base": {
                "has_const": use_constant,
                "tokens": operators
            }
        },
        stage_config={
            "default": {
                "operators": operators,
                "time_limit": time_limit,
                "n_psrn_inputs": n_inputs,
                "n_sample_variables": 3,
            },
            "stages": [
                {},
            ],
        },
    )

start = time.time()
    flag, pareto_ls = regressor.fit(
        Input,
        Output,
        n_down_sample=n_down_sample,
        use_threshold=False,
        threshold=1e-20,
        probe=probe,
        prun_const=True,
        prun_ndigit=6,
        top_k=topk,
    )
    end = time.time()
    time_cost = end - start

pareto_ls = regressor.display_expr_table(sort_by='mse') # or 'reward'

expr_str, reward, loss, complexity = pareto_ls[0]
    print('Found:', expr_str, 'time_cost', time_cost)

```

</details>

## 📚 Citation

If you use this work, please cite:

```bibtex
@article{Ruan2026Discovering,
  title     = {Discovering physical laws with parallel symbolic enumeration},
  author    = {Ruan, Kai and Xu, Yilong and Gao, Ze-Feng and Liu, Yang and Guo, Yike and Wen, Ji-Rong and Sun, Hao},
  journal   = {Nature Computational Science},
  volume    = {6},
  number    = {1},
  pages     = {53--66},
  year      = {2026},
  month     = {jan},
  publisher = {Springer Nature},
  doi       = {10.1038/s43588-025-00904-8},
  url       = {https://doi.org/10.1038/s43588-025-00904-8}
}
```

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