AI-Assisted Tumor-Growth Law Discovery from Sparse Data: Refitting and LLM-Based Ranking

Yitian Lu; Weinan Wang

cs.LGOct 12, 2026

AI-Assisted Tumor-Growth Law Discovery from Sparse Data: Refitting and LLM-Based Ranking

Authors: Yitian Lu, Weinan Wang

Organizations: Department of Mathematics, University of Oklahoma, Norman, OK, USA.

Abstract

Inferring tumor-growth laws from sparse, noisy observations requires more than accurate curve fitting. We study an AI-assisted workflow involving symbolic regression, mathematical admissibility checks, coefficient refitting, and large language model (LLM) ranking. Synthetic Gompertz data allow trajectory fit, forecasting, and recovery of the governing growth law to be assessed separately. A dense noiseless control recovers the generating family, but sparse noisy sampling frequently leads to alternative forms. Among retained candidates in the tested configurations, physics-informed neural network refitting often increases growth-law error, whereas classical nonlinear least squares usually reduces it. A separate exploratory analysis of archived LLM rankings shows no consistent improvement over selection by training error and identifies errors in mathematical reasoning. Plausible saturation, accurate trajectory fits, and convincing explanations can therefore coexist with incorrect growth laws. Mathematical validity and law recovery require evaluation alongside predictive performance.

Figures & tables

Figure 1. Representative trajectories with $K=20$ and $V_{0}=0.1$ . Gompertz uses $a=0.5$ ; the generalized-logistic curves use $\nu=0.5,1,2$ and $r=0.5/\nu$ in the Richards convention above. Sparse uniform sampling can concentrate observations near saturation, where several laws are difficult to distinguish.

Figure 2. A candidate law can have a stable positive equilibrium and still be inadmissible from the observed initial condition. The forward-positivity filter rejects candidates that drive $V(t)$ negative or leave the candidate’s domain on the evaluation interval.

Figure 3. Dense noiseless positive control with 50 observations on $[0,14]$ . The curve is the solved PINN-refitted ODE. The rounded recovered expression $V(1.50-0.500\log V)$ is algebraically equivalent to Gompertz with $a=0.5$ and $K=e^{3}\approx 20.09$ .

Figure 4. A separate sparse example with $N=6$ , $\sigma=0.08$ , data seed 47, and observations on $[0,10]$ . The PySR law $V(0.7568282-0.037870117V)$ follows the estimated phase-space relation but has a different trajectory from Gompertz truth. The blue Gompertz curve is fitted to spline-derived $(V,\dot{V})$ pairs, not to trajectory residuals. Trajectories are displayed through $t=12.5$ .

Quantity	Value
Truth	Gompertz, $a=0.5,K=20,V_{0}=0.1$
Data regime	$N=6,\ \sigma=0.08,\ R=30$
Default PySR choice is non-Gompertz	$29/30=97\%$ $[83\%,99\%]$
Lowest training-RMSE candidate is Gompertz	$3/30=10\%$ $[3\%,26\%]$
Generating law has lower training RMSE than default choice	$24/30=80\%$ $[63\%,90\%]$
Median training RMSE, generating law	$1.145$

Table 1. Ranking instability across 30 sparse noisy Gompertz realizations. Trajectory errors use the six noisy training observations; the generating-law baseline uses the known parameters. Bracketed intervals are two-sided 95% Wilson intervals. The 24/30 comparison counts two failed candidate integrations as infinite error; the corresponding finite-only count is 22/28.

Method	Training MSE	Forecast RMSE	$E_{f}^{\mathrm{full}}$
PySR constants	2.302	2.417	0.198
PINN-refit ODE	4.008	2.850	0.261
Classical NLS ODE	0.987	1.823	0.378

Table 2. Seed-45 summary for a structure-preserving quadratic candidate

f(V)=V(0.2526-0.01562V)+2.1931

from a sparse noisy Gompertz benchmark (seed 45,

N=6

\sigma=0.20

\lambda_{\mathrm{phys}}=0.1

). The PINN representative uses initialization seed 53, selected from 20 starts by training MSE. Smaller values are better. The column

E_{f}^{\mathrm{full}}

denotes the full law-level error in ( 4.3 ).

Method	Training MSE	Forecast RMSE	$E_{f}^{\mathrm{full}}$
PySR ODE	13.284	2.199	0.427
PINN-refit ODE	13.473	4.245	0.434
PINN network output	12.802	2.690	–
Classical NLS ODE	12.781	3.532	0.400

Table 3. Seed-47 summary for the linear candidate

f(V)=3.288-0.1394V

from seed 47,

N=6

\sigma=0.20

, and

\lambda_{\mathrm{phys}}=0.1

. The PINN representative uses initialization seed 55, selected from 20 starts by training MSE. The PINN network output and the solved PINN-refit ODE are distinct objects.

$\lambda_{\mathrm{phys}}$	Training-trajectory MSE	Forecast RMSE	$E_{f}^{\mathrm{full}}$
0.1	3.8852	1.5939	0.3910
1	6.9876	3.2325	0.4545
10	9.9859	2.3051	0.4709

Table 4. Medians over selected PINN representatives for

\sigma=5\%,10\%,15\%,20\%

N=6

, and data seed 47. Each expression contributes its retained start with the smallest training MSE. The retained candidate counts are 26/44, 29/44, and 28/44 for physics weights 0.1, 1, and 10, respectively; 36 of the 44 original expressions pass the initial filter at each weight. Candidate expressions and repeated starts are not independent data realizations.

Figure 5. A noiseless Richards example with $\nu=2$ , $r=0.5$ , $K=20$ , $V_{0}=0.1$ , and 12 observations on $[0,11]$ , fitted with a Gompertz family. The forecast RMSE of 15.238 uses 200 times on $[11,15]$ . This illustration is distinct from the Gompertz-generated comparisons in Table 5 .

Regime	$N$	$\sigma$	Finite/total	Forecast RMSE		Lower error
				Fixed Gompertz	Candidate median
Sparse	12	0.05	7/11	0.214	0.325	0/7
Medium	20	0.03	8/11	0.146	0.308	2/8
Dense, $k=3$	40	0.01	8/11	0.112	0.204	1/8
Dense, $k=5$	40	0.01	4/12	0.112	0.078	4/4

Table 5. Gompertz-generated comparisons at data seed 47. The fixed-family error is from one fit; the candidate error is a median over finite PINN-refitted forecasts. “Lower error” counts candidate expressions that outperform the fixed-Gompertz fit. All rows except the last use spline degree

k=3

$\sigma$	$n$	Selected ID		Forecast RMSE		$E_{f}^{\mathrm{full}}$
		LLM	Train	LLM	Train	LLM	Train
0.00	24	C6	C21	0.139	0.123	0.330	0.123
0.05	26	C9	C17	0.289	0.526	0.177	0.091
0.10	27	C12	C12	0.815	0.815	0.235	0.235
0.15	20	C9	C18	2.125	1.296	0.346	0.329
0.20	27	C15	C24	3.532	1.549	0.400	0.355

Table 6. Retrospective scoring of the six archived LLM rankings.

n

is the pool size; “Train” selects the smallest supplied training MSE. IDs are local to each pool. Both forecast RMSE and law error are computed on the exact historical expressions, not the later numerical refits.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Data seed	Metric	Mean	SD	Median	IQR	Min	Max
45	Training MSE	5.745	0.919	5.864	0.964	4.008	7.740
45	Forecast RMSE	3.111	0.096	3.140	0.085	2.850	3.279
45	$E_{f}^{\mathrm{full}}$	0.293	0.015	0.296	0.014	0.261	0.321
47	Training MSE	13.811	0.373	13.715	0.284	13.473	14.834
47	Forecast RMSE	4.447	0.223	4.419	0.206	4.064	5.047
47	$E_{f}^{\mathrm{full}}$	0.443	0.008	0.441	0.007	0.434	0.464

Appendix

Table 7. Across-start PINN-refit ODE errors for the two

20\%

-noise examples. Each row contains 20 valid starts; values are rounded to three decimals.

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