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Getting started: discover a PDE from field data

KD recovers the Burgers equation from a bundled benchmark dataset whose equation is known. This notebook loads the data, inspects the field, fits a symbolic model in one call, compares the result with the known equation, and plots the convergence curve.

import kd

ds = kd.load_burgers()
print(ds.ground_truth)

u_t = -u * u_x + 0.1 * u_xx

Look at the data

The dataset contains a field u sampled on an (x, t) grid. Before fitting, kd.preview prints the grid, the field, and the left-hand-side derivative that KD will use.

kd.preview(ds)
kd.viz.plots.plot_field(ds)

Dataset: burgers
Axes:
     x | n=256  | range [-8.000, 7.938] | step 0.0625 (uniform)
     t | n=201  | range [0.000, 10.000] | step 0.05 (uniform)
Fields:
     u | dtype=float64 | shape=(256, 201) | min=-1.000 max=1.000 mean=0.000 (NaN=0)
LHS: u_t  (field='u', axis='t')
Status: ready to fit

png

Fit a model

SGA-PDE searches over symbolic expression trees and scores candidate right-hand sides against the target derivative. generations sets the search budget. The value below is small enough to run quickly and large enough to recover a Burgers-like equation.

model = kd.Model(algorithm="sga", generations=5, seed=0, verbose=False).fit(ds)
print(model.result_.equation)
print(model.best_score_)

u_t = -u*u_x + 0.1002*u_xx
-28.776750720652373

Compare to ground truth

The discovered equation is KD's best symbolic right-hand side for u_t. The exact string can differ from the compact textbook form, so the check is whether it contains the nonlinear advection and diffusion terms.

print(f"Discovered  : {model.result_.equation}")
print(f"Ground truth: {ds.ground_truth}")

Discovered  : u_t = -u*u_x + 0.1002*u_xx
Ground truth: u_t = -u * u_x + 0.1 * u_xx

Visualize

plot_convergence draws the recorded best-score curve, labels it with the algorithm's score kind (AIC for SGA), and notes when the best score was reached on the first iteration. It creates its own figure; the returned list contains any plot notes.

from kd.viz.plots import plot_convergence

plot_convergence(model.result_);

png

The report

model.report writes every supported figure and an HTML report, and displays the report here. The same files remain in the output directory.

model.report("../out/getting_started")

Report: ../out/getting_started/report.html

Next steps

This notebook used SGA-PDE. KD provides seven algorithms behind the same kd.Model interface: