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PySR: High-Performance Symbolic Regression in Python

PySR uses evolutionary algorithms to search for symbolic expressions which optimize a particular objective.

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(pronounced like py as in python, and then sur as in surface)

If you find PySR useful, please cite it using the citation information given in CITATION.md. If you've finished a project with PySR, please submit a PR to showcase your work on the Research Showcase page!

Test status

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Linux Windows macOS
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PySR is built on an extremely optimized pure-Julia backend: SymbolicRegression.jl.

Symbolic regression is a very interpretable machine learning algorithm for low-dimensional problems: these tools search equation space to find algebraic relations that approximate a dataset.

One can also extend these approaches to higher-dimensional spaces by using a neural network as proxy, as explained in 2006.11287, where we apply it to N-body problems. Here, one essentially uses symbolic regression to convert a neural net to an analytic equation. Thus, these tools simultaneously present an explicit and powerful way to interpret deep models.

Backstory:

Previously, we have used eureqa, which is a very efficient and user-friendly tool. However, eureqa is GUI-only, doesn't allow for user-defined operators, has no distributed capabilities, and has become proprietary (and recently been merged into an online service). Thus, the goal of this package is to have an open-source symbolic regression tool as efficient as eureqa, while also exposing a configurable python interface.

Installation

pip - recommended
(works everywhere)
conda
(Linux and Intel-based macOS)
docker
(if all else fails)
1. Install Julia
2. Then, run: pip install -U pysr
3. Finally, to install Julia packages:
python -c 'import pysr; pysr.install()'
conda install -c conda-forge pysr 1. Clone this repo.
2. docker build -t pysr .
Run with:
docker run -it --rm pysr ipython

Common issues tend to be related to Python not finding Julia. To debug this, try running python -c 'import os; print(os.environ["PATH"])'. If none of these folders contain your Julia binary, then you need to add Julia's bin folder to your PATH environment variable.

Running PySR on macOS with an M1 processor: you should use the pip version, and make sure to get the Julia binary for ARM/M-series processors.

Introduction

Let's create a PySR example. First, let's import numpy to generate some test data:

import numpy as np

X = 2 * np.random.randn(100, 5)
y = 2.5382 * np.cos(X[:, 3]) + X[:, 0] ** 2 - 0.5
We have created a dataset with 100 datapoints, with 5 features each. The relation we wish to model is \(2.5382 \cos(x_3) + x_0^2 - 0.5\).

Now, let's create a PySR model and train it. PySR's main interface is in the style of scikit-learn:

from pysr import PySRRegressor

model = PySRRegressor(
    model_selection="best",  # Result is mix of simplicity+accuracy
    niterations=40,
    binary_operators=["+", "*"],
    unary_operators=[
        "cos",
        "exp",
        "sin",
        "inv(x) = 1/x",
    # ^ Custom operator (julia syntax)
    ],
    extra_sympy_mappings={"inv": lambda x: 1 / x},
    # ^ Define operator for SymPy as well
    loss="loss(x, y) = (x - y)^2",
    # ^ Custom loss function (julia syntax)
)
This will set up the model for 40 iterations of the search code, which contains hundreds of thousands of mutations and equation evaluations.

Let's train this model on our dataset:

model.fit(X, y)
Internally, this launches a Julia process which will do a multithreaded search for equations to fit the dataset.

Equations will be printed during training, and once you are satisfied, you may quit early by hitting 'q' and then \<enter>.

After the model has been fit, you can run model.predict(X) to see the predictions on a given dataset.

You may run:

print(model)
to print the learned equations:
PySRRegressor.equations_ = [
       pick     score                                           equation       loss  complexity
    0        0.000000                                          4.4324794  42.354317           1
    1        1.255691                                          (x0 * x0)   3.437307           3
    2        0.011629                          ((x0 * x0) + -0.28087974)   3.358285           5
    3        0.897855                              ((x0 * x0) + cos(x3))   1.368308           6
    4        0.857018                ((x0 * x0) + (cos(x3) * 2.4566472))   0.246483           8
    5  >>>>       inf  (((cos(x3) + -0.19699033) * 2.5382123) + (x0 *...   0.000000          10
]
This arrow in the pick column indicates which equation is currently selected by your model_selection strategy for prediction. (You may change model_selection after .fit(X, y) as well.)

model.equations_ is a pandas DataFrame containing all equations, including callable format (lambda_format), SymPy format (sympy_format - which you can also get with model.sympy()), and even JAX and PyTorch format (both of which are differentiable - which you can get with model.jax() and model.pytorch()).

Note that PySRRegressor stores the state of the last search, and will restart from where you left off the next time you call .fit(), assuming you have set warm_start=True. This will cause problems if significant changes are made to the search parameters (like changing the operators). You can run model.reset() to reset the state.

You will notice that PySR will save two files: hall_of_fame...csv and hall_of_fame...pkl. The csv file is a list of equations and their losses, and the pkl file is a saved state of the model. You may load the model from the pkl file with:

model = PySRRegressor.from_file("hall_of_fame.2022-08-10_100832.281.pkl")

There are several other useful features such as denoising (e.g., denoising=True), feature selection (e.g., select_k_features=3). For examples of these and other features, see the examples page. For a detailed look at more options, see the options page. You can also see the full API at this page.

Detailed Example

The following code makes use of as many PySR features as possible. Note that is just a demonstration of features and you should not use this example as-is. For details on what each parameter does, check out the API page.

model = PySRRegressor(
    procs=4,
    populations=8,
    # ^ 2 populations per core, so one is always running.
    population_size=50,
    ncyclesperiteration=500, 
    # ^ Generations between migrations.
    niterations=10000000,  # Run forever
    early_stop_condition=(
        "stop_if(loss, complexity) = loss < 1e-6 && complexity < 10"
        # Stop early if we find a good and simple equation
    ),
    timeout_in_seconds=60 * 60 * 24,
    # ^ Alternatively, stop after 24 hours have passed.
    maxsize=50,
    # ^ Allow greater complexity.
    maxdepth=10,
    # ^ But, avoid deep nesting.
    binary_operators=["*", "+", "-", "/"],
    unary_operators=["square", "cube", "exp", "cos2(x)=cos(x)^2"],
    constraints={
        "/": (-1, 9),
        "square": 9,
        "cube": 9,
        "exp": 9,
    },
    # ^ Limit the complexity within each argument.
    # "inv": (-1, 9) states that the numerator has no constraint,
    # but the denominator has a max complexity of 9.
    # "exp": 9 simply states that `exp` can only have
    # an expression of complexity 9 as input.
    nested_constraints={
        "square": {"square": 1, "cube": 1, "exp": 0},
        "cube": {"square": 1, "cube": 1, "exp": 0},
        "exp": {"square": 1, "cube": 1, "exp": 0},
    },
    # ^ Nesting constraints on operators. For example,
    # "square(exp(x))" is not allowed, since "square": {"exp": 0}.
    complexity_of_operators={"/": 2, "exp": 3},
    # ^ Custom complexity of particular operators.
    complexity_of_constants=2,
    # ^ Punish constants more than variables
    select_k_features=4,
    # ^ Train on only the 4 most important features
    progress=True,
    # ^ Can set to false if printing to a file.
    weight_randomize=0.1,
    # ^ Randomize the tree much more frequently
    cluster_manager=None,
    # ^ Can be set to, e.g., "slurm", to run a slurm
    # cluster. Just launch one script from the head node.
    precision=64,
    # ^ Higher precision calculations.
    warm_start=True,
    # ^ Start from where left off.
    turbo=True,
    # ^ Faster evaluation (experimental)
    julia_project=None,
    # ^ Can set to the path of a folder containing the
    # "SymbolicRegression.jl" repo, for custom modifications.
    update=False,
    # ^ Don't update Julia packages
    extra_sympy_mappings={"cos2": lambda x: sympy.cos(x)**2},
    # extra_torch_mappings={sympy.cos: torch.cos},
    # ^ Not needed as cos already defined, but this
    # is how you define custom torch operators.
    # extra_jax_mappings={sympy.cos: "jnp.cos"},
    # ^ For JAX, one passes a string.
)

Docker

You can also test out PySR in Docker, without installing it locally, by running the following command in the root directory of this repo:

docker build -t pysr .
This builds an image called pysr for your system's architecture, which also contains IPython.

You can then run this with:

docker run -it --rm -v "$PWD:/data" pysr ipython
which will link the current directory to the container's /data directory and then launch ipython.

If you have issues building for your system's architecture, you can emulate another architecture by including --platform linux/amd64, before the build and run commands.