Column chromatography

This page fits two algorithms, DISCOVER and PySR, to the same dataset and compares the expressions they return. Both are called through kd.Model, so the two runs differ in the algorithm name and in the search parameters specific to each algorithm.

The dataset

kd.load_tlc_cc() returns 74 rows from an automated column chromatography platform. Each row is one experimental condition: the two feature columns are the thin-layer retardation factor R_F and the petroleum-ether fraction r of the eluent, and the target is the retention volume V_S in millilitres, averaged over the compounds run at that condition. Xu et al. (2025) published a formula relating the three:

\[V_S = \dfrac{r}{0.147\,R_F + 0.0114}\]

Run both algorithms

table = kd.load_tlc_cc()

discover = kd.Model(
    algorithm="discover", generations=200, seed=0,
    batch_size=1000, max_length=15, reward_alpha=0.005,
).fit(table)

pysr = kd.Model(
    algorithm="pysr", generations=100, seed=0,
    maxsize=10, binary_operators=("+", "*", "/"), unary_operators=(),
).fit(table)

Neither call specifies candidate terms. DISCOVER searches over the table's own column names together with the operators add, sub, mul and div. PySR receives the same two columns and the operators +, * and /. Both return a REGRESSION equation on result_.equation and the nondominated set of the search on result_.pareto_front().

Pareto front of each algorithm, plotted on a common accuracy axis: R² over all 74 rows against the size of the expression. The two size axes are not the same measure, since PySR counts nodes and DISCOVER counts tokens. The dashed line marks the fit of the published formula on these rows, 0.876. Circles mark the front entry equal to that formula, and diamonds mark the answer each run selected.

The published formula on both fronts

Each algorithm returns an expression together with an outer coefficient fitted for it. Dividing through puts both in the form of the published formula:

	a	b	R²
Published	0.147	0.0114	0.876
DISCOVER, 5 tokens	0.1507	0.01110	0.876
PySR, 7 nodes	0.1496	0.01121	0.876

The two coefficient pairs are within 2.6% of the published values and within 1% of each other, from one run of each algorithm at a fixed seed.

The selected answers

Neither run selects the published formula. Three of the 74 rows were measured at r = 0, where that formula predicts zero against measured volumes of 4.41, 5.17 and 5.86 mL, so an expression carrying an intercept fits the table more closely, and a shorter expression that omits r is cheaper under both complexity measures. DISCOVER selected a nine-token expression with an R² of 0.898, and PySR selected the five-node expression 5.892 / (R_F + 0.0669) with an R² of 0.864. The published formula appears between these on both fronts.