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equation discovery · symbolic regression

KD A platform for Knowledge Discovery

KD is a plug-in platform for equation discovery. It gives different discovery algorithms a consistent, easy-to-use interface and provides shared services such as visualization, evaluation, and experiment management, while leaving each algorithm free to implement its own search procedure.

Quick start User guide Examples
SGA-PDE · kdv · seed 42 generation 200 / 200 · nmse 2.70e-06 · AIC -19.42
Heatmap of the kdv field u over x and t
\(u_t =\)\(-u\,u_x\)\(-0.0025\,u_{xxx}\)
An SGA-PDE run on the KdV dataset, showing the field it was given and the equation it returned.

Quick start

Install KD and run your first equation-discovery experiment in a few lines of Python. The example below uses a Burgers dataset bundled with KD. Follow Installation for Python environment setup, then activate the environment before running the code below.

1  install
python -m pip install sail-kd

uv venv
uv pip install sail-kd --torch-backend=auto
2  fit
import kd
# a 256 × 201 (x, t) field; for your own arrays, PDEDataset.from_arrays
data = kd.load_burgers()

model = kd.Model(algorithm="sga", generations=5, seed=0)
model.fit(data)
5 generations · AIC -28.78
\(u_t =\)\(-u\,u_x\)\(+0.1002\,u_{xx}\)

Experimental controller

After installing the agent extra and configuring an endpoint:

kd-agent run --data sample.npz "Find the equation governing this data."

This experimental controller uses a language model to select and run KD algorithms, inspect their results and write a search report. See the agent example for file and bundled-dataset commands, or the controller guide for configuration and interactive use.

Examples

notebook
Getting started
Load a bundled benchmark, fit in one call, read the result.
notebook
Use your own data
Two coordinate arrays and one field array, from NumPy to a fitted equation.
notebookSGA
Walkthrough
A 200-generation SGA-PDE fit on the KdV benchmark, read term by term, with every figure and the report.
kd-agent run \
--data sample.npz \
"Find the governing equation."
command lineexperimental
Agent search
Run a short search, save a checkpoint and continue from it, with the results of both segments.
notebookEqGPT
Breaking waves
The published EqGPT equation, reproduced on 12 wave-tank experiments.
notebookDLGA
Noisy Burgers walkthrough
The Burgers benchmark with 15% Gaussian noise added, recovered by DLGA from a network fitted to the noisy field.
worked exampleDISCOVERPySR
Column chromatography
Two algorithms on one 74-row table, both recovering the published formula.

Algorithms

KD currently provides seven equation-discovery algorithms through a common interface.

AICmediumfinite_diff autograd
SGA-PDE
A genetic search over expression trees; the coefficients come from the algorithm's own sparse regression.
DLGA fitnessheavyautograd
DLGA-PDE
A neural network fits the field and supplies derivatives; a genetic algorithm searches term sets over a candidate library.
rewardheavyfinite_diff autograd
DISCOVER
A recurrent controller writes expression trees token by token and is trained on its own best-rewarded draws; the coefficients come from a platform refit.
EqGPT rewardheavyfinite_diff
EqGPT
A pretrained transformer proposes equations and is fine-tuned on candidates ranked by their fit to data.
LLM4ED sparse rewardmediumfinite_diff
LLM4ED
A large language model proposes and revises candidate equations round by round, filtered by a sparse reward.
NMSEmediumfinite_diffexternal package
PySR
Multi-population evolutionary search returning a complexity/error Pareto front.
NMSElightfinite_diffexternal package
PySINDy
Sparse regression over a supplied candidate library, solved in one pass.

Design

design
Platform and plugins
The responsibilities shared by the platform and those implemented by each algorithm.
architecture
Architecture
The complete component diagram, import layers and execution order of a fit.
data
Data requirements
Fields, coordinate axes, grid or scattered points, and how the left-hand side is chosen.
score
Score conventions and comparability
Each algorithm builds its own final score. Refit the terms in one shared context before comparing.