MOSAIC-SR: Transformer-Guided Symbolic Regression for Scientific Equation Recovery

Peiyi Zheng; Yanming Kang; Hans De Sterck; Giang Tran

cs.LGSep 17, 2026

MOSAIC-SR: Transformer-Guided Symbolic Regression for Scientific Equation Recovery

Authors: Peiyi Zheng, Yanming Kang, Hans De Sterck, Giang Tran

Organizations: University of Waterloo

Abstract

Symbolic regression aims to recover closed-form equations from observations, providing interpretable models for scientific discovery. Existing approaches struggle to combine flexible structural search with efficient inference. Search-based methods can refine expression structure but often rely on costly combinatorial optimization with random initialization. Pretrained neural models generate formulas almost instantly, but their predictions often contain symbolic errors. We introduce MOSAIC-SR, which uses a pretrained Transformer to propose multiple initial sketches. These sketches initialize searches in several promising regions, avoiding random starts in the vast expression space. Each search jointly recovers structure and constants through scale-aware constant optimization and local symbolic repair. We evaluate MOSAIC-SR on the SRSD-Feynman dataset with and without dummy variables and on six additional benchmarks. MOSAIC-SR obtains the highest symbolic solution rate on every dataset while ranking among the top two methods in predictive accuracy. This advantage persists in the presence of irrelevant dummy inputs. The results show that learned priors can focus search on promising equation structures, and that numerical optimization and symbolic repair are important for recovery.

Figures & tables

Figure 1: Overview of MOSAIC-SR. Stage 1 aligns numerical observations with symbolic expressions through contrastive pretraining, and Stage 2 trains the decoder to predict operator structures with variable and constant placeholders. At inference time, MCTS maps these variable tokens to observed inputs, and the resulting equations initialize parallel search trajectories that alternate constant fitting with structural repair. The final expression is selected according to validation NMSE and complexity.

Cited from SRSD-Feynman
		Easy (30)		Medium (40)		Hard (50)
Method	Type	Sym	Acc	Sym	Acc	Sym	Acc
uDSR	RL	50.0	100.0	17.5	75.0	4.0	20.0
PySR	GP	60.0	66.7	30.0	45.0	4.0	38.0
DSR	RL	46.7	63.3	10.0	45.0	2.0	28.0
AI Feynman	Rules	30.0	33.3	2.5	5.0	2.0	6.0

Table 1: Results on SRSD-Feynman. Easy, Medium, and Hard contain 30, 40, and 50 equations. Results cited from SRSD-Feynman are taken directly from the original paper. Results evaluated in this work are means over five seeds. Sym is the symbolic solution rate in percent, and Acc is the percentage of equations with

R^{2}>0.999

. The top three values per column, including ties, are bolded. The best value in each column is also italicized.

Cited from SRSD-Feynman
		Easy (30)		Medium (40)		Hard (50)
Method	Type	Sym	Acc	Sym	Acc	Sym	Acc
uDSR	RL	10.0	53.3	7.5	37.5	0.0	12.0
PySR	GP	20.0	20.0	5.0	10.0	0.0	2.0
DSR	RL	10.0	76.7	0.0	45.0	2.0	22.0
AI Feynman	Rules	0.0	6.7	0.0	0.0	0.0	0.0

Table 2: Results on SRSD-Feynman with dummy variables. Results cited from SRSD-Feynman are taken directly from the original paper. Results evaluated in this work are means over five seeds. Sym is the symbolic solution rate in percent, and Acc is the percentage of problems with test

R^{2}>0.999

. The top three values in each column, including ties, are bolded. The best value in each column is also italicized.

Method	Nguyen			Korns			Keijzer			Vladislavleva			Strogatz			Livermore
	Sym	Acc	Time	Sym	Acc	Time	Sym	Acc	Time	Sym	Acc	Time	Sym	Acc	Time	Sym	Acc	Time
PySR	8.3	91.7	99.6	20.0	60.0	619.6	26.7	73.3	100.8	12.5	25.0	126.9	42.9	71.4	150.6	31.8	72.7	146.2
Operon	0.0	0.0	4.6	6.7	53.3	20.4	0.0	20.0	6.5	0.0	12.5	11.7	7.1	21.4	9.7	0.0	0.0	6.5
GP-GOMEA	8.3	16.7	29.2	0.0	26.7	88.9	6.7	13.3	33.3	12.5	12.5	53.2	14.3	64.3	45.9	18.2	18.2	31.4
E2E-SR	0.0	0.0	9.0	0.0	6.7	21.9	0.0	0.0	13.4	0.0	0.0	18.2	0.0	0.0	13.6	0.0	0.0	12.4
SNIP-SR	0.0	0.0	26.3	6.7	6.7	171.9	0.0	0.0	25.1	0.0	0.0	101.9	0.0	7.1	25.4	0.0	0.0	24.6

Table 3: Results and runtime on six symbolic regression benchmarks. Sym and Acc are percentages. Time is the average runtime in seconds per equation. Acc denotes test NMSE below

10^{-6}

. For each dataset, the top three Sym and Acc values are bolded.

Appendix figures & tables39 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 2: Predictive accuracy and symbolic solution rate on SRSD-Feynman without dummy variables. Points show mean performance on the Easy, Medium, and Hard splits. Horizontal bars show the reported variation across seeds.

Panel of Figure 3 — Figure 3: Rank-based comparisons on SRSD-Feynman without dummy variables. The panels compare mean predictive accuracy rank with symbolic solution rank and expression complexity rank, respectively. Methods closer to the lower left corner achieve better tradeoffs.

Figure 4: Predictive accuracy and symbolic solution rate on SRSD-Feynman with dummy variables. Points show mean performance on the Easy, Medium, and Hard splits. Horizontal bars show the reported variation across seeds.

Panel of Figure 5 — Figure 5: Rank based comparisons on SRSD-Feynman with dummy variables. The panels compare mean predictive accuracy rank with symbolic solution rank and expression complexity rank, respectively. Methods closer to the lower left corner achieve better tradeoffs.

Figure 6: Best validation NMSE versus wall-clock time on SRSD-Feynman Easy, grouped by ground-truth operator count. The 150 runs comprise 30 equations evaluated over five seeds. The groups contain one or two, three, and four true operators, with 40, 55, and 55.

Figure 7: Best validation NMSE versus wall-clock time on SRSD-Feynman Easy, grouped by top-1 decoded sketch operator count. The runs are divided into sketches with fewer than 10, between 10 and 15, and more than 15 operators, with 57, 44, and 49 runs respectively.

Figure 8: Best validation NMSE versus wall-clock time on SRSD-Feynman Medium, grouped by ground-truth operator count. The 200 runs comprise 40 equations evaluated over five seeds. The groups contain fewer than five, five or six, and more than six true operators, with 75, 95, and 30 runs, respectively.

Figure 9: Best validation NMSE versus wall-clock time on SRSD-Feynman Medium, grouped by top-1 decoded sketch operator count. The groups contain fewer than 13, between 13 and 17, and more than 17 decoded operators, with 82, 53, and 65 runs respectively.

Figure 10: Best validation NMSE versus wall-clock time during the 100-second search window on SRSD-Feynman Hard, grouped by ground-truth operator count. The 250 runs comprise 50 equations evaluated over five seeds. The groups contain fewer than nine, between nine and 11, and more than 11 true operators, with 115, 75, and 60 runs, respectively.

Figure 11: Best validation NMSE versus wall-clock time on SRSD-Feynman Hard, grouped by top-1 decoded sketch operator count. The x-axis spans 0–100 seconds. The 250 runs comprise 50 equations evaluated over five seeds. The groups contain fewer than 13, between 13 and 19, and more than 19 decoded operators, with 107, 73, and 70 runs, respectively.

	Decoder		Decoder + MCTS		Full MOSAIC
Seed	Sym	Acc	Sym	Acc	Sym	Acc
0	36.7	60.0	43.3	66.7	83.3	96.7
1	30.0	46.7	30.0	60.0	83.3	96.7
2	30.0	46.7	40.0	66.7	90.0	100.0

Appendix

Table 4: Component ablation on 30 clean SRSD-Feynman Easy equations. Sym and Acc report symbolic and numerical recovery. All values are percentages. Decoder, Decoder + MCTS, and Full MOSAIC are evaluated with the same three method seeds.

	100 steps		300 steps		500 steps
Paths	Sym	Acc	Sym	Acc	Sym	Acc
1	47.8	70.0	53.3	76.7	56.7	80.0
6	57.8	77.8	70.0	87.8	75.6	90.0
12	66.7	83.3	76.7	93.3	82.2	94.4
24	70.0	84.4	85.6	97.8	85.6	97.8
48	73.3	87.8	85.6	98.9	87.8	98.9

Appendix

Table 5: Search budget ablation on 30 SRSD-Feynman Easy equations. Sym and Acc denote symbolic and numerical recovery percentages averaged over three method seeds.

Setting	Stage 1	Stage 2
Batch size	$128$	$128$
Training steps	$400{,}000$	$250{,}000$
Peak learning rate	$10^{-4}$	$10^{-5}$
Encoder learning rate	$10^{-4}$	$5\times 10^{-6}$
Warmup steps	$16{,}000$	$2{,}000$
Learning-rate schedule	Cosine decay	Constant

Appendix

Table 6: Core optimization settings used in the two pretraining stages.

Parameter	Value
Set-Transformer numeric encoder
Architecture	Set Transformer
Input dimension	11 (10 variables + output)
ISAB blocks	4
Hidden dimension	256
Attention heads	8

Appendix

Table 7: Detailed Transformer architecture used by MOSAIC-SR, including the numeric encoder, symbolic encoder, and symbolic decoder. The complete architecture has approximately

17.4

million parameters and is relatively lightweight.

Parameter	Value
Independent search paths	24
Mutations per path	300
Decoder/search passes	6
Decoder sketches retained	24
Workers	12
Conditional MCTS	Enabled

Appendix

Table 8: MOSAIC-SR inference time decoder, MCTS, and local-search budget for the cross-benchmark experiments.

Group	Problems	Variables	1/2/3 dummy	Train/Val/Test
Without dummy variables		Input domain extrema: $10^{-30}$ – $10^{4}$
Easy	30	1–5	0/0/0	$8{,}000/1{,}000/1{,}000$
Medium	40	1–6	0/0/0	$8{,}000/1{,}000/1{,}000$
Hard	50	1–8	0/0/0	$8{,}000/1{,}000/1{,}000$
With dummy variables		Input domain extrema: $10^{-33}$ – $10^{33}$
Easy + Dummy	30	3–8	6/11/13	$8{,}000/1{,}000/1{,}000$

Appendix

Table 9: SRSD-Feynman benchmark groups, variable count ranges, dummy variable distribution, data split sizes, and input domain extrema [ 17 ] . The extrema summarize equation specific sampling distributions rather than a shared interval. Dummy counts give the numbers of problems containing one, two, or three irrelevant variables.

Dataset	Problems	Variables	Training rows	Test rows
Nguyen	12	1–2	20	20
Korns	15	5	$10{,}000$	$10{,}000$
Keijzer	15	1–3	20– $1{,}000$	120– $361{,}201$
Vladislavleva	8	1–5	30– $1{,}024$	221– $93{,}636$
Strogatz	14	2	300	100
Livermore	22	1–2	20– $1{,}000$	20– $1{,}000$

Appendix

Table 10: Numbers of problems, variable count ranges, training and test sample counts for the Nguyen [ 29 ] , Korns [ 11 ] , Keijzer [ 10 ] , Vladislavleva [ 35 ] , Strogatz [ 27 , 3 ] , and Livermore [ 21 ] benchmarks.

Taylor data	2	$\displaystyle x_{0}\allowbreak+\frac{(-1)\allowbreak\cdot x_{0}^{2}}{2}$
	5	$\displaystyle\frac{x_{0}^{5}}{5}\allowbreak-\frac{x_{0}^{4}}{4}\allowbreak+\frac{x_{0}^{3}}{3}\allowbreak-\frac{x_{0}^{2}}{2}\allowbreak+x_{0}$
	7	$\displaystyle\frac{x_{0}^{7}}{7}\allowbreak-\frac{x_{0}^{6}}{6}\allowbreak+\frac{x_{0}^{5}}{5}\allowbreak-\frac{x_{0}^{4}}{4}\allowbreak+\frac{x_{0}^{3}}{3}\allowbreak-\frac{x_{0}^{2}}{2}\allowbreak+x_{0}$
	10	$\displaystyle\frac{x_{0}^{9}}{9}\allowbreak-\frac{x_{0}^{8}}{8}\allowbreak+\frac{x_{0}^{7}}{7}\allowbreak-\frac{x_{0}^{6}}{6}\allowbreak+\frac{x_{0}^{5}}{5}\allowbreak-\frac{x_{0}^{4}}{4}\allowbreak+\frac{x_{0}^{3}}{3}\allowbreak-\frac{x_{0}^{2}}{2}\allowbreak+x_{0}\allowbreak+\frac{(-1)\allowbreak\cdot x_{0}^{10}}{10}$
MOSAIC-SR	2	$\displaystyle-0.5\allowbreak\cdot x_{0}^{2}\allowbreak+x_{0}$
	5	$\displaystyle\frac{x_{0}\allowbreak\cdot(0.6\allowbreak\cdot x_{0}^{4}\allowbreak-0.75\allowbreak\cdot x_{0}^{3}\allowbreak+x_{0}^{2}\allowbreak-1.5\allowbreak\cdot x_{0}\allowbreak+3.0)}{3}$
	7	$\displaystyle\log{(x_{0}\allowbreak+1.00006603079797)}$

Appendix

Table 22

Taylor data	2	$\displaystyle x_{0}$
	5	$\displaystyle\frac{x_{0}^{5}}{120}\allowbreak-\frac{x_{0}^{3}}{6}\allowbreak+x_{0}$
	7	$\displaystyle\frac{x_{0}^{5}}{120}\allowbreak-\frac{x_{0}^{3}}{6}\allowbreak+x_{0}\allowbreak+\frac{(-1)\allowbreak\cdot x_{0}^{7}}{5040}$
	10	$\displaystyle\frac{x_{0}^{9}}{362880}\allowbreak-\frac{x_{0}^{7}}{5040}\allowbreak+\frac{x_{0}^{5}}{120}\allowbreak-\frac{x_{0}^{3}}{6}\allowbreak+x_{0}$
MOSAIC-SR	2	$\displaystyle x_{0}$
	5	$\displaystyle-0.15\allowbreak\cdot x_{0}^{3}\allowbreak-0.00833333333333333\allowbreak\cdot x_{0}\allowbreak\cdot(-(x_{0}^{2}\allowbreak-1)^{2}\allowbreak-0.949359216081706)\allowbreak+0.983755339865986\allowbreak\cdot x_{0}$
	7	$\displaystyle 0.995212385036962\allowbreak\cdot\sin{(x_{0})}$

Appendix

Table 23

Taylor data	2	$\displaystyle-x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}\allowbreak+\pi\allowbreak\cdot x_{0}\allowbreak\cdot x_{1}$
	5	$\displaystyle\ldots\allowbreak+\frac{-x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{5}}{120}\allowbreak+\frac{\pi\allowbreak\cdot x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{4}}{24}\allowbreak-\frac{\pi^{2}\allowbreak\cdot x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{3}}{12}\allowbreak+\frac{x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{3}}{6}\allowbreak-\frac{\pi\allowbreak\cdot x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{2}}{2}\allowbreak+\frac{\pi^{3}\allowbreak\cdot x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{2}}{12}$
	7	$\displaystyle\ldots\allowbreak+\frac{x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{7}}{5040}\allowbreak-\frac{\pi\allowbreak\cdot x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{6}}{720}\allowbreak-\frac{x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{5}}{120}\allowbreak+\frac{\pi^{2}\allowbreak\cdot x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{5}}{240}\allowbreak-\frac{\pi^{3}\allowbreak\cdot x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{4}}{144}\allowbreak+\pi\allowbreak\cdot x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{4}$
	10	$\displaystyle\ldots\allowbreak+\frac{-x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{9}}{362880}\allowbreak+\frac{\pi\allowbreak\cdot x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{8}}{40320}\allowbreak-\frac{\pi^{2}\allowbreak\cdot x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{7}}{10080}\allowbreak+\frac{x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{7}}{5040}\allowbreak-\frac{\pi\allowbreak\cdot x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}^{6}}{720}\allowbreak+\pi^{3}\allowbreak\cdot x_{0}\allowbreak\cdot x_{1}$
MOSAIC-SR	2	$\displaystyle x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot(\pi\allowbreak-x_{2})$
	5	$\displaystyle x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot\sin{(x_{2})}$
	7	$\displaystyle x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot\sin{(1.002337079977\allowbreak\cdot x_{2})}$

Appendix

Table 24

Taylor data	2	$\displaystyle x_{0}\allowbreak\cdot x_{1}\allowbreak-\frac{\sqrt{2}\allowbreak\cdot x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}\allowbreak\cdot x_{4}^{2}}{4}\allowbreak+\frac{\sqrt{2}\allowbreak\cdot\pi\allowbreak\cdot x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}\allowbreak\cdot x_{4}}{8}\allowbreak+\frac{\sqrt{2}\allowbreak\cdot x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}\allowbreak\cdot x_{4}}{2}\allowbreak-\frac{\sqrt{2}\allowbreak\cdot\pi\allowbreak\cdot x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}}{8}\allowbreak-\frac{\sqrt{2}\allowbreak\cdot\pi^{2}\allowbreak\cdot x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}}{64}\allowbreak+\frac{\sqrt{2}\allowbreak\cdot x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}}{2}$
	5	$\displaystyle\ldots\allowbreak+x_{0}\allowbreak\cdot x_{1}\allowbreak+\frac{\sqrt{2}\allowbreak\cdot x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}\allowbreak\cdot x_{4}^{5}}{240}\allowbreak-\frac{\sqrt{2}\allowbreak\cdot\pi\allowbreak\cdot x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}\allowbreak\cdot x_{4}^{4}}{192}\allowbreak+\frac{\sqrt{2}\allowbreak\cdot x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}\allowbreak\cdot x_{4}^{4}}{48}\allowbreak-\sqrt{2}\allowbreak\cdot x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}\allowbreak\cdot x_{4}$
	7	$\displaystyle\ldots\allowbreak+x_{0}\allowbreak\cdot x_{1}\allowbreak-\frac{\sqrt{2}\allowbreak\cdot x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}\allowbreak\cdot x_{4}^{7}}{10080}\allowbreak-\frac{\sqrt{2}\allowbreak\cdot x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}\allowbreak\cdot x_{4}^{6}}{1440}\allowbreak+\frac{\sqrt{2}\allowbreak\cdot\pi\allowbreak\cdot x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}\allowbreak\cdot x_{4}^{6}}{5760}\allowbreak-\sqrt{2}\allowbreak\cdot\pi^{2}\allowbreak\cdot x_{0}$
	10	$\displaystyle\ldots\allowbreak+x_{0}\allowbreak\cdot x_{1}\allowbreak-\frac{\sqrt{2}\allowbreak\cdot x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}\allowbreak\cdot x_{4}^{10}}{7257600}\allowbreak+\frac{\sqrt{2}\allowbreak\cdot\pi\allowbreak\cdot x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}\allowbreak\cdot x_{4}^{9}}{2903040}\allowbreak+\frac{\sqrt{2}\allowbreak\cdot x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}\allowbreak\cdot x_{4}^{9}}{725760}\allowbreak-\sqrt{2}\allowbreak\cdot\pi\allowbreak\cdot x_{0}$
MOSAIC-SR	2	$\displaystyle x_{0}\allowbreak\cdot(x_{1}\allowbreak+0.712793490396519\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}\allowbreak\cdot x_{4}\allowbreak+0.0669510918992685\allowbreak\cdot x_{2}\allowbreak\cdot x_{3})$
	5	$\displaystyle x_{0}\allowbreak\cdot(x_{1}\allowbreak+x_{2}\allowbreak\cdot x_{3}\allowbreak\cdot\sin{(x_{4})})$
	7	$\displaystyle x_{0}\allowbreak\cdot(x_{1}\allowbreak+1.0\allowbreak\cdot x_{2}\allowbreak\cdot x_{3}\allowbreak\cdot\sin{(x_{4})})$

Appendix

Table 25

Method	Prediction
MOSAIC-SR	$x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}\allowbreak\cdot\sin(x_{3})$
PySR	$x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}\allowbreak\cdot((x_{2}\allowbreak-(x_{3}\allowbreak+(x_{3}\allowbreak+\sin{(x_{2})}\allowbreak+\cos{(x_{3})})\allowbreak\cdot\cos{(x_{1})}))\allowbreak\cdot(-1.090031\allowbreak\cdot 10^{-8})\allowbreak+\sin{(x_{3})})$
Operon	$1.0\allowbreak\cdot\sin{(1.0\allowbreak\cdot x_{3})}\allowbreak\cdot(-0.0027)\allowbreak\cdot x_{1}\allowbreak\cdot(0.0\allowbreak\cdot x_{2}\allowbreak\cdot e^{\sin{(\cos{(-2.0643\allowbreak\cdot x_{3})})}}\allowbreak-2.8718\allowbreak\cdot x_{2})\allowbreak\cdot(-1.1801\allowbreak\cdot\ldots\allowbreak\cdot x_{1}\allowbreak+127.7102\allowbreak\cdot x_{0})\allowbreak+0.0$
GP-GOMEA	$0.560146\allowbreak\cdot x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}\allowbreak\cdot e^{\cos{(\sin{(45.244)})}}\allowbreak\cdot\sin{(x_{3})}\allowbreak+0.0$
E2E-SR	$(-0.00047451032853731212\allowbreak\cdot x_{1}\allowbreak-19.400000000000002\allowbreak\cdot(\ldots\allowbreak-0.0034)\allowbreak\cdot(30.38912823447775\allowbreak\cdot x_{0}\allowbreak+28.70121528936986)^{2})\allowbreak\cdot(\sin{(0.5475682868652938\allowbreak\cdot x_{3}\allowbreak-1.7094145799484706)}\allowbreak-0.0010999999999999998)$
SNIP-SR	$(2.9242714060991837\allowbreak\cdot x_{0}\allowbreak+2.8138086776705413\allowbreak\cdot x_{2})\allowbreak\cdot(0.058021734319863794\allowbreak\cdot x_{1}\allowbreak-(-0.11031702200615587)\allowbreak\cdot 1\allowbreak\cdot\frac{1}{\ldots\allowbreak-1.980797909\allowbreak-0.6172368400935637})\allowbreak-0.3353196824463989$

Appendix

Table 26

Method	Prediction
MOSAIC-SR	$\sqrt{\frac{x_{0}\allowbreak\cdot x_{1}}{x_{2}}}$
PySR	$\sqrt{\frac{x_{0}\allowbreak\cdot x_{1}}{x_{2}}}$
Operon	$0.0\allowbreak+1.0\allowbreak\cdot(-0.3697\allowbreak\cdot x_{0}\allowbreak+37267.543\allowbreak\cdot x_{1}\allowbreak-0.3693\allowbreak\cdot x_{2}\allowbreak+\log{(-(-1)\allowbreak\cdot 405434.875\allowbreak\cdot x_{1}\allowbreak+0.0036\allowbreak\cdot x_{2}\allowbreak+1.3023)}\allowbreak+0.2423)\allowbreak\cdot(-1.5741)\allowbreak\cdot\frac{1}{\ldots\allowbreak-1.316}$
GP-GOMEA	$0.088311\allowbreak\cdot\sqrt{x_{0}\allowbreak\cdot x_{1}}\allowbreak\cdot(\frac{x_{2}}{451.691759865387}\allowbreak+\sin{(x_{2})}\allowbreak-3.832)\allowbreak\cdot(-x_{1}\allowbreak+x_{2}\allowbreak-8.635\allowbreak+3.845)\allowbreak-2.6\allowbreak\cdot 10^{-5}$
E2E-SR	$(0.3\allowbreak+\frac{0.3}{0.35927349724435038\allowbreak\cdot x_{2}\allowbreak+0.1556208603753693})\allowbreak\cdot(\ldots\allowbreak+0.00019999999999999998\allowbreak\cdot\sqrt{(77.563144512331082\allowbreak\cdot x_{0}\allowbreak-25.889254950623197)\allowbreak\cdot(369968.1560669523\allowbreak\cdot x_{1}\allowbreak-0.683865090989673)})$
SNIP-SR	$0.0008867575457021422\allowbreak\cdot x_{0}\allowbreak-(0.007603756123524418\allowbreak\cdot x_{1}\allowbreak-0.013302360696511688\allowbreak\cdot x_{2}\allowbreak-(\ldots\allowbreak-0.0017)\allowbreak+0.0617609181319026\allowbreak+0.07176091798886611)\allowbreak+0.0042390819407224665$

Appendix

Table 27

Method	Prediction
MOSAIC-SR	$\frac{2\allowbreak\cdot\pi\allowbreak\cdot x_{0}}{x_{1}\allowbreak\cdot x_{2}}$
PySR	$x_{0}\allowbreak\cdot 6.2831855\allowbreak\cdot\frac{1}{x_{1}\allowbreak\cdot x_{2}}$
Operon	$1.0\allowbreak\cdot(-1.72)\allowbreak\cdot x_{0}\allowbreak\cdot\frac{1}{(-1)\allowbreak\cdot 0.5005\allowbreak\cdot x_{1}}\allowbreak\cdot\frac{1}{0.5469\allowbreak\cdot x_{2}}\allowbreak+354.8659$
GP-GOMEA	$10326321737.098848\allowbreak\cdot(x_{0}\allowbreak+x_{0}\allowbreak-(x_{0}\allowbreak+x_{1})\allowbreak\cdot\sin{(x_{0})}\allowbreak-(-x_{1}\allowbreak+x_{2}))\allowbreak\cdot e^{-\sqrt{x_{1}}\allowbreak+\cos{(x_{1})}}\allowbreak-715320458.443097$
E2E-SR	$-0.00046093254802904855\allowbreak\cdot x_{0}\allowbreak+(23.8\allowbreak-0.03485279359938367\allowbreak\cdot e^{0.03610003915333401\allowbreak\cdot x_{1}})\allowbreak\cdot(0.016461876715323162\allowbreak\cdot x_{0}\allowbreak+0.004137440901804627)\allowbreak\cdot(\ldots\allowbreak-3165117.571990305\allowbreak\cdot x_{2}\allowbreak-0.03078025829268)$
SNIP-SR	$x_{0}\allowbreak\cdot e^{17.656406591853308\allowbreak-0.29635039365231586\allowbreak\cdot x_{1}}\allowbreak\cdot 1\allowbreak\cdot\frac{1}{5611389.50330818\allowbreak\cdot x_{2}\allowbreak+6.00837704285586\allowbreak\cdot 10^{-5}}\allowbreak-607824857.6181431$

Appendix

Table 28

Method	Prediction
MOSAIC-SR	$\frac{x_{0}\allowbreak\cdot x_{3}\allowbreak\cdot(x_{1}\allowbreak-x_{2})}{x_{4}}$
PySR	$\displaystyle\frac{-1.1901392\allowbreak\cdot 0.84023786\allowbreak\cdot x_{0}\allowbreak\cdot x_{3}\allowbreak\cdot(-x_{1}\allowbreak+x_{2})}{x_{4}}$
Operon	$0.0\allowbreak+1.0\allowbreak\cdot(0.0\allowbreak\cdot x_{1}\allowbreak+\frac{(-0.0018\allowbreak\cdot x_{1}\allowbreak+0.0018\allowbreak\cdot x_{2})\allowbreak\cdot(-1.1077)\allowbreak\cdot x_{0}\allowbreak\cdot(30998.9414\allowbreak\cdot x_{3}\allowbreak+\ldots)}{63.3968\allowbreak\cdot x_{4}})$
GP-GOMEA	$-0.338812\allowbreak-0.204571\allowbreak\cdot\sqrt{x_{3}}\allowbreak\cdot x_{0}^{2}\allowbreak\cdot(x_{4}\allowbreak\cdot x_{1}\allowbreak-(x_{1}\allowbreak-x_{2}))\allowbreak\cdot e^{\sin{x_{2}}\allowbreak-x_{4}\allowbreak\cdot x_{0}}$
E2E-SR	$(\ldots\allowbreak+0.006059999999999999)\allowbreak\cdot 14.7\allowbreak\cdot\lvert\ldots\rvert\allowbreak\cdot(-6.38)\allowbreak\cdot(0.0003498200466886168\allowbreak\cdot x_{0}\allowbreak+0.0001793022405506989)^{2}$
SNIP-SR	$0.2680922469550011\allowbreak+\frac{(\ldots\allowbreak-2.0815424010112733\allowbreak\cdot x_{2}\allowbreak+2.07626\allowbreak\cdot x_{1})}{(0.752254473241603\allowbreak-0.7920820084794776\allowbreak\cdot x_{4})\allowbreak\cdot(21.622118920174636\allowbreak\cdot x_{4}\allowbreak-0.02949544189126491)}$

Appendix

Table 29

Method	Prediction
MOSAIC-SR	$\sqrt{(x_{0}\allowbreak-x_{1})^{2}\allowbreak+(x_{2}\allowbreak-x_{3})^{2}}$
PySR	$\sqrt{(x_{0}\allowbreak-x_{1})(x_{0}\allowbreak-x_{1}\allowbreak-2.5\allowbreak\cdot 10^{-7})\allowbreak+(x_{2}\allowbreak-x_{3})^{2}}$
Operon	$1.0\allowbreak\cdot(\ldots\allowbreak+0.1434\allowbreak\cdot x_{0}\allowbreak\cdot(-0.7572)\allowbreak\cdot x_{1}\allowbreak+0.759\allowbreak\cdot x_{0}\allowbreak-0.1087\allowbreak\cdot x_{2}\allowbreak\cdot(-0.6665)\allowbreak\cdot x_{2}\allowbreak+(0.0003\allowbreak\cdot x_{2}\allowbreak-0.0002\allowbreak\cdot x_{3})\allowbreak\cdot(-347.5512)\allowbreak\cdot x_{3})\allowbreak+0.0$
GP-GOMEA	$0.770164\allowbreak\cdot(\sqrt{x_{2}\allowbreak-x_{3}}\allowbreak\cdot\sqrt{x_{2}\allowbreak-x_{3}}\allowbreak+\sqrt{(-x_{0}\allowbreak+x_{1})\allowbreak\cdot(-x_{0}\allowbreak+x_{1})})\allowbreak+0.315539$
E2E-SR	$0.07540000000000001\allowbreak\cdot\ldots\allowbreak+(-1.272562287287044\allowbreak\cdot x_{0}\allowbreak-0.0780069168581643)\allowbreak\cdot(0.00078562116483634423\allowbreak\cdot x_{3}\allowbreak+0.00902697873784341)\allowbreak+3.52\allowbreak\cdot\lvert 0.25922662262108978\allowbreak\cdot x_{3}\allowbreak+0.03979767739670666\rvert$
SNIP-SR	$0.17291808796873243\allowbreak\cdot\lvert 4.53620345462416\allowbreak\cdot x_{2}\allowbreak-4.550165321708825\allowbreak\cdot x_{3}\rvert\allowbreak+0.10160093303248278\allowbreak\cdot\lvert\ldots\allowbreak-7.43758068232461\allowbreak\cdot x_{0}\allowbreak+7.587149571342\rvert\allowbreak+0.237128160071409$

Appendix

Table 30

Method	Prediction
MOSAIC-SR	$\frac{1.607\allowbreak\cdot 10^{26}\allowbreak\cdot x_{0}}{x_{1}^{2}}\allowbreak+1.788\allowbreak\cdot 10^{9}\allowbreak\cdot x_{2}^{2}$
PySR	$(x_{0}\allowbreak+x_{2}\allowbreak\cdot x_{2})\allowbreak\cdot 1828542300.0$
Operon	$1.0\allowbreak\cdot(-192.0\allowbreak\cdot x_{0}\allowbreak+904458403840.0\allowbreak\cdot(\ldots\allowbreak+247344122675455525388288.0\allowbreak\cdot x_{2}))\allowbreak-2901952.75$
GP-GOMEA	$0.551209\allowbreak\cdot(-\ldots\allowbreak\cdot x_{0}\allowbreak+x_{2}\allowbreak\cdot x_{2}\allowbreak\cdot(x_{0}\allowbreak\cdot x_{2}\allowbreak-3393026213.036)\allowbreak\cdot\sin{(11963926169.822001)})\allowbreak+51687513017.491699$
E2E-SR	$(9.074947371606654\allowbreak-3.9595126882746026\allowbreak\cdot 10^{-11}\allowbreak\cdot x_{1})\allowbreak\cdot(0.00019487679974378762\allowbreak\cdot x_{0}\allowbreak+89.997268492792118)\allowbreak\cdot(2.6396751255164016\allowbreak\cdot 10^{-11}\allowbreak\cdot x_{1}\allowbreak+89.950035085595564)\allowbreak\cdot 0.8\allowbreak\cdot(\ldots\allowbreak-0.0\allowbreak+55.505828465024757)$
SNIP-SR	$1.6002061046745069\allowbreak\cdot(7.741215005617892\allowbreak-85.30817533148172\allowbreak\cdot x_{2})^{2}\allowbreak-0.09952661536012392\allowbreak\cdot(\ldots\allowbreak-9.581409582884245\allowbreak\cdot x_{0}\allowbreak+2279.3307399561345)\allowbreak+5962.31522397963$

Appendix

Table 31

Method	Prediction
MOSAIC-SR	$\frac{x_{0}\allowbreak\cdot(x_{1}^{2}\allowbreak+x_{2}^{2}\allowbreak+x_{3}^{2})}{2}$
PySR	$(x_{2}\allowbreak\cdot x_{2}\allowbreak-(x_{1}\allowbreak\cdot(2.6849122\allowbreak\cdot 10^{-8}\allowbreak-x_{1})\allowbreak-x_{3}\allowbreak\cdot x_{3}))\allowbreak\cdot 0.594442\allowbreak\cdot x_{0}\allowbreak\cdot 0.84112495$
Operon	$1.0\allowbreak\cdot(-\ldots\allowbreak+0.0044\allowbreak\cdot x_{0}\allowbreak\cdot 0.8345\allowbreak\cdot x_{2}\allowbreak\cdot(0.0\allowbreak\cdot x_{0}\allowbreak+135.0603\allowbreak\cdot x_{2}\allowbreak-0.0\allowbreak\cdot x_{3})\allowbreak+0.002\allowbreak\cdot x_{3}\allowbreak\cdot(-8.7617)\allowbreak\cdot x_{3}\allowbreak\cdot(-29.1342)\allowbreak\cdot x_{0})\allowbreak+0.0$
GP-GOMEA	$-0.466588\allowbreak\cdot x_{0}\allowbreak\cdot((-x_{1}\allowbreak+x_{2})\allowbreak\cdot(x_{1}\allowbreak-x_{2})\allowbreak-(x_{1}\allowbreak\cdot x_{2}\allowbreak+x_{3}\allowbreak\cdot x_{3}))\allowbreak+1.557416$
E2E-SR	$0.060000000000000005\allowbreak\cdot\lvert(0.25330816576971352\allowbreak\cdot x_{0}\allowbreak+0.04483942013849232)\allowbreak\cdot(0.060000000000000005\allowbreak\cdot(\ldots\allowbreak+22.0195174746776)\allowbreak+0.060000000000000005\allowbreak\cdot(19.7499472505697\allowbreak\cdot x_{1}\allowbreak+2.85561506727142)^{2})\rvert$
SNIP-SR	$(0.01840704821112982\allowbreak\cdot x_{0}\allowbreak+0.027442044563726835)^{2}\allowbreak\cdot(9.879745390847708\allowbreak\cdot x_{3}\allowbreak-6.194380210972868\allowbreak\cdot 1\allowbreak\cdot\frac{1}{\ldots\allowbreak-4.9351334859321385\allowbreak+0.0072800592259153})\allowbreak+13.64280530393845$

Appendix

Table 32

Method	Prediction
MOSAIC-SR	$\frac{1.380649\allowbreak\cdot 10^{-23}\allowbreak\cdot x_{1}}{x_{2}\allowbreak\cdot(x_{0}\allowbreak-1)}$
PySR	$e^{\frac{\cos{(\sin{(0.733933372794484\allowbreak\cdot x_{1})})}}{\cos{(5.7083836\allowbreak\cdot\sin{x_{0}})}}}\allowbreak\cdot\cos{(\sin{(\log{x_{1}})})}$
Operon	$24.9136\allowbreak+0.9928\allowbreak\cdot\frac{e^{\sin{(6.2437\allowbreak\cdot x_{1})}\allowbreak+0.9405\allowbreak\cdot x_{1}\allowbreak-10.8365}}{(\sin{(4.6620\allowbreak\cdot x_{1})}\allowbreak+1)\allowbreak\cdot(\sqrt{0.9405\allowbreak\cdot x_{1}}\allowbreak+\ldots)}$
GP-GOMEA	$24.874483\allowbreak-1.5238670125519831\allowbreak\cdot 10^{97}\allowbreak\cdot e^{(x_{2}\allowbreak-x_{0})\allowbreak\cdot(220.661\allowbreak\cdot x_{0})}\allowbreak\cdot(\sqrt{x_{1}}\allowbreak+\sqrt{-11502.059}\allowbreak+\ldots)$
E2E-SR	$0.00783\allowbreak\cdot\left\lvert(20.5\allowbreak-\frac{0.635}{(0.01955351496819132\allowbreak\cdot x_{0}\allowbreak-0.0195847291063135)\allowbreak\cdot(1.49301283178395\allowbreak\cdot 10^{19}\allowbreak\cdot x_{2}\allowbreak-0.0018505074099962075)})\allowbreak\cdot\ldots\right\rvert$
SNIP-SR	$-12.202334840601555\allowbreak-0.0035234845524787085\allowbreak\cdot\frac{(5.868107132362321\allowbreak-6.00000000000062\allowbreak\cdot x_{2})^{2}}{-1.4130094715317487\allowbreak+1.4542114898814111\allowbreak\cdot x_{1}}\allowbreak+\ldots$

Appendix

Table 33

Method	Prediction
MOSAIC-SR	$\frac{0.3989\allowbreak\cdot\exp\!\left(-0.5\allowbreak\cdot\frac{x_{0}^{2}}{x_{1}^{2}}\right)}{x_{1}}$
PySR	$\frac{\sin{(\sin{(x_{0}\allowbreak\cdot 0.44023314\allowbreak\cdot\frac{1}{x_{1}})}\allowbreak\cdot 3.1941698)}}{x_{0}\allowbreak\cdot(x_{0}\allowbreak\cdot x_{0}\allowbreak\cdot 0.55104643\allowbreak\cdot\frac{1}{x_{1}\allowbreak\cdot x_{1}}\allowbreak+3.5161734)}$
Operon	$0.933\allowbreak\cdot(-e^{e^{-\frac{0.6815\allowbreak\cdot x_{1}\allowbreak-\cos{(-0.9175\allowbreak\cdot x_{0})}}{\cos{(\ldots)}}\allowbreak+\cos{(-0.418\allowbreak\cdot x_{0})}}}\allowbreak+e^{e^{-(0.6795\allowbreak\cdot x_{1}\allowbreak-\cos{(-0.9167\allowbreak\cdot x_{0})})\allowbreak+\cos{(-0.4191\allowbreak\cdot x_{0})}}})\allowbreak+0.0543$
GP-GOMEA	$0.0\allowbreak\cdot e^{-(\sqrt{x_{0}}\allowbreak+x_{1}\allowbreak-26.453)\allowbreak+\sqrt{-47.357}\allowbreak\cdot\cos{(x_{0})}}\allowbreak+0.041035$
E2E-SR	$\ldots\allowbreak+1.282299505658052\allowbreak\cdot 10^{-8}\allowbreak\cdot(-0.049681955299757749\allowbreak\cdot x_{0}\allowbreak-0.82316389700632219)^{2}\allowbreak\cdot(3.8724298635018757\allowbreak\cdot(-\operatorname{atan}{(18.363084125663076\allowbreak\cdot x_{0}\allowbreak-30.50603247242241)})\allowbreak-8.06163517864935)^{2}\allowbreak-1.7384408$
SNIP-SR	$0.03971435283239767\allowbreak+e^{1.1335725598924378\allowbreak-0.5147795206145964\allowbreak\cdot x_{1}}\allowbreak\cdot 1\allowbreak\cdot\frac{1}{10.003562082456531\allowbreak-0.03732680107658152\allowbreak\cdot x_{0}}$

Appendix

Table 34

Method	Prediction
MOSAIC-SR	$\frac{769089.72\allowbreak\cdot x_{0}}{\sqrt{\frac{x_{1}\allowbreak+x_{2}}{x_{0}}}}$
PySR	$769089.702160698\allowbreak\cdot x_{0}\allowbreak\cdot\sqrt{\frac{x_{0}}{x_{1}\allowbreak+x_{2}}}$
Operon	$0.9991\allowbreak\cdot(-0.0153\allowbreak\cdot\ldots\allowbreak+(-0.0018\allowbreak\cdot x_{0}\allowbreak+\frac{0.1102\allowbreak\cdot x_{0}}{\log{(-124481448181760.0\allowbreak\cdot x_{0}\allowbreak+1.3068\allowbreak\cdot x_{1}\allowbreak+1.3515\allowbreak\cdot x_{2})}})\allowbreak\cdot\sqrt{\sqrt{3.1423\allowbreak\cdot x_{0}}})\allowbreak+194935.9375$
GP-GOMEA	$0.0\allowbreak\cdot\sqrt{x_{0}}\allowbreak\cdot x_{0}\allowbreak\cdot(27792532242127954164842496.0\allowbreak-x_{0})\allowbreak\cdot(-\sqrt{x_{0}\allowbreak--15960121508062442932928512.0}\allowbreak+\sqrt{x_{1}\allowbreak+x_{2}})\allowbreak+2592428.592193$
E2E-SR	$(9.750140310322155\allowbreak-3.5192830562550441\allowbreak\cdot 10^{-10}\allowbreak\cdot x_{0})\allowbreak\cdot 8.41\allowbreak\cdot(-2.77\allowbreak+\frac{0.04650000000000001}{-8.5274752133162408\allowbreak\cdot 10^{-27}\allowbreak\cdot x_{1}\allowbreak-6.8927613619355\allowbreak\cdot 10^{-27}\allowbreak\cdot x_{2}\allowbreak-0.007543467070585526})^{2}\allowbreak\cdot(\ldots\allowbreak+2.4049168319)$
SNIP-SR	–

Appendix

Table 35

Method	Prediction
MOSAIC-SR	$\frac{0.5\allowbreak\cdot x_{1}^{2}}{x_{0}}$
PySR	$x_{2}\allowbreak\cdot(-213.32457)\allowbreak\cdot\frac{1}{x_{4}\allowbreak-1\allowbreak\cdot 0.75216883}$
Operon	$1.0\allowbreak\cdot(\ldots\allowbreak+\frac{187.625}{e^{40126340.0\allowbreak\cdot x_{1}}}\allowbreak+\frac{102002000.0}{e^{32823758.0\allowbreak\cdot x_{1}}})\allowbreak-358206272.0$
GP-GOMEA	$8235444610619720859648.0\allowbreak-4117722181658013597696.0\allowbreak\cdot(\sqrt{\sqrt{x_{0}}}\allowbreak+\cos{(\sqrt{x_{1}})}\allowbreak+\cos{(x_{1}\allowbreak\cdot x_{1}\allowbreak\cdot(x_{2}\allowbreak-64871391778.814003))})$
E2E-SR	$83.3\allowbreak\cdot(-\ldots)\allowbreak\cdot(2996997476.088138\allowbreak\cdot x_{1}\allowbreak+(0.0027957553159015011\allowbreak-128292.52353780801\allowbreak\cdot x_{1})\allowbreak\cdot(-886458768.06676007\allowbreak\cdot x_{3}\allowbreak-4344590016.9748102\allowbreak\cdot x_{5}\allowbreak-79.82124690760698)\allowbreak+0.700142210497718)^{2}$
SNIP-SR	$2475.4420995751925\allowbreak\cdot(785.1938480983365\allowbreak\cdot x_{1}\allowbreak\cdot(\ldots\allowbreak+501.62243037789665\allowbreak\cdot(248.55715087494292\allowbreak-9.9\allowbreak\cdot x_{0})^{3})\allowbreak+456.80185093394306)\allowbreak+380.8454602832127$

Appendix

Table 36

Method	Prediction
MOSAIC-SR	$-\frac{x_{0}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}}{x_{3}}$
PySR	$\displaystyle\frac{x_{0}\allowbreak\cdot x_{2}\allowbreak\cdot(116.94989\allowbreak-x_{1})}{x_{3}}$
Operon	$-1556.0109\allowbreak+2.5895843701950685\allowbreak\cdot 10^{25}\allowbreak\cdot\frac{2.4524}{(-2.7636805717985786\allowbreak\cdot 10^{27}\allowbreak\cdot x_{3})\allowbreak\cdot x_{1}^{-1}\allowbreak\cdot(-25.7193)\allowbreak\cdot x_{2}\allowbreak\cdot(-0.2950)\allowbreak\cdot x_{0}}$
GP-GOMEA	$-2029461039.160109\allowbreak-2.4147765954346173\allowbreak\cdot 10^{46}\allowbreak\cdot\cos{(\cos{x_{1}}\allowbreak\cdot(x_{2}\allowbreak-76913126293.195007))}\allowbreak\cdot x_{1}^{2}\allowbreak\cdot(x_{0}\allowbreak+x_{3})^{2}\allowbreak\cdot\sin{x_{2}}$
E2E-SR	$(66.829576481052452\allowbreak\cdot x_{2}\allowbreak-0.0017371407519285152)\allowbreak\cdot(-92.6)\allowbreak\cdot(0.1078068308644714\allowbreak-3.2338505696933849\allowbreak\cdot 10^{-11}\allowbreak\cdot x_{1})^{2}\allowbreak\cdot(4.390271757193469\allowbreak\cdot 10^{27}\allowbreak\cdot x_{0}\allowbreak-0.3531506873096935)^{2}\allowbreak\cdot\ldots$
SNIP-SR	$753690240.1795971\allowbreak+x_{2}\allowbreak\cdot(4.368785986955585\allowbreak\cdot x_{1}\allowbreak-43.75053176204654\allowbreak\cdot e^{17.15960871844588\allowbreak-2.0\allowbreak\cdot x_{0}\allowbreak\cdot x_{3}})\allowbreak\cdot\tan{(-7.954465726265568\allowbreak+73364.55500241455\allowbreak\cdot\ldots)}$

Appendix

Table 37

Method	Prediction
MOSAIC-SR	$1.0\allowbreak\cdot x_{1}\allowbreak\cdot(x_{1}^{3}\allowbreak+1)\allowbreak\cdot(1.0\allowbreak\cdot x_{1}\allowbreak\cdot(x_{1}\allowbreak+1)\allowbreak+1)$
PySR	$(x_{1}\allowbreak+1.0)\allowbreak\cdot(x_{1}\allowbreak\cdot x_{1}\allowbreak\cdot x_{1}\allowbreak\cdot(x_{1}\allowbreak\cdot x_{1}\allowbreak+1.0)\allowbreak-x_{1}\allowbreak+x_{1}\allowbreak+x_{1}\allowbreak-1\allowbreak\cdot 3.009093\allowbreak\cdot 10^{-8})$
Operon	$0.263197660446167\allowbreak\cdot\exp(3.121897220611572\allowbreak\cdot x_{1})\allowbreak-0.250046372413635$
GP-GOMEA	$0.780175498806059$
E2E-SR	$(0.82722131708105334\allowbreak-0.0089588507630547431\allowbreak\cdot x_{1})\allowbreak\cdot(-0.014978078619482149\allowbreak\cdot x_{1}\allowbreak-4.659629985505114)\allowbreak\cdot(0.06345852623830444\allowbreak\cdot x_{1}\allowbreak+0.06393233734253951)\allowbreak\cdot(1.5553560352525597\allowbreak\cdot x_{1}\allowbreak+0.00177689565047806)\allowbreak\cdot(0.0137\allowbreak\cdot(-6.4184099828749757\allowbreak\cdot x_{1}\allowbreak-0.6423212964360453)^{3}\allowbreak-2.63)$
SNIP-SR	$-787.69917995555147\allowbreak\cdot(0.00070707667540883934\allowbreak\cdot x_{1}\allowbreak+1)^{2}\allowbreak+784.8060091216709\allowbreak-289.476805583656\allowbreak\cdot(73.38150650502881\allowbreak\cdot x_{1}\allowbreak-101.72796954182427)^{-1}$

Appendix

Table 38

Method	Prediction
MOSAIC-SR	$\exp((1.0\allowbreak\cdot x_{2}\allowbreak+1.11259407539204\allowbreak\cdot 10^{-17})\allowbreak\cdot\log(x_{1}))$
PySR	$\exp(x_{2}\allowbreak\cdot\log(x_{1}\allowbreak-7.453925\allowbreak\cdot 10^{-9}))\allowbreak+9.259128\allowbreak\cdot 10^{-9}$
Operon	$0.4424457312568320138828227\allowbreak\cdot x_{1}\allowbreak+0.526753962039948$
GP-GOMEA	$2.01164555549622\allowbreak-1.81258428096771\allowbreak\cdot\cos(2\allowbreak\cdot x_{2}\allowbreak\cdot(x_{1}^{2}\allowbreak-1.00295400619507)\allowbreak\cdot(5.04862117767334\allowbreak\cdot x_{1}\allowbreak+x_{2}\allowbreak+\cos(x_{2})\allowbreak-0.310173720121384)^{-1}\allowbreak+0.978409945964813)$
E2E-SR	$0.62014723269920278\allowbreak\cdot x_{1}\allowbreak+(0.052757803417621763\allowbreak-0.054718873473459068\allowbreak\cdot x_{1})\allowbreak\cdot(0.11515042468674191\allowbreak-0.031915295185792375\allowbreak\cdot x_{2})\allowbreak\cdot(94.508205254545958\allowbreak-203.30043033349743\allowbreak\cdot x_{2})\allowbreak+0.4187448946002867$
SNIP-SR	$1.22898636248581\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}\allowbreak-1.1248163129914415\allowbreak\cdot x_{2}\allowbreak+0.9561923734505865$

Appendix

Table 39

Method	Prediction
MOSAIC-SR	$(0.9\allowbreak\cdot\cos(x_{2})^{2}\allowbreak+0.0999999999999999)\allowbreak\cdot\sin(x_{1})$
PySR	$\sin(x_{1})\allowbreak\cdot(\cos(\sin(x_{2})\allowbreak\cdot 11.785839^{-1})\allowbreak-0.99600285)\allowbreak\cdot 250.22122$
Operon	$0.05662523819582111\allowbreak\cdot x_{2}\allowbreak+0.017341127619147$
GP-GOMEA	$0.450004816055298\allowbreak\cdot\sin(x_{1})\allowbreak\cdot\sin(2\allowbreak\cdot x_{2}\allowbreak-4.71237802505493)\allowbreak+0.550020277500153\allowbreak\cdot\sin(x_{1})\allowbreak-7.75317403167719\allowbreak\cdot 10^{-6}$
E2E-SR	$0.03790008096684952\allowbreak\cdot\sin(0.09445851906577567\allowbreak+0.03960031064088332\allowbreak\cdot((0.05920507672980812\allowbreak-90.0083517060878\allowbreak\cdot\tan(0.49632673547196763\allowbreak\cdot x_{1}\allowbreak+0.22256771097376267))\allowbreak\cdot(0.2089712876837641\allowbreak\cdot x_{2}\allowbreak-0.900448620172386))^{-1})\allowbreak-0.02225101908746781$
SNIP-SR	$0.013779661168824415\allowbreak\cdot x_{1}^{2}\allowbreak+0.008255847908988898\allowbreak\cdot x_{1}\allowbreak-0.13407144869680693$

Appendix

Table 40

Method	Prediction
MOSAIC-SR	$x_{1}\allowbreak+\sin(x_{1}^{2})\allowbreak+0.333333333333333$
PySR	$x_{1}\allowbreak+\sin(x_{1}\allowbreak\cdot x_{1})\allowbreak+0.3333334$
Operon	$0.99815390177276\allowbreak\cdot x_{1}\allowbreak+0.385755509138107$
GP-GOMEA	$0.416014617587274$
E2E-SR	$0.99510452368622162\allowbreak\cdot x_{1}\allowbreak-1.03\allowbreak\cdot\sin(0.777\allowbreak\cdot(1.1888631901860294\allowbreak\cdot x_{1}\allowbreak+0.090269873934594466)^{2}\allowbreak-3.77)\allowbreak+0.36129693740020308$
SNIP-SR	$-90.25067542494121\allowbreak\cdot(0.005531892247014312\allowbreak\cdot x_{1}\allowbreak-1)^{2}\allowbreak+90.72787811926341\allowbreak+0.0022187878284230704\allowbreak\cdot(0.14762793267964414\allowbreak\cdot x_{1}\allowbreak+1.4559888306142952)^{-1}$

Appendix

Table 41

Method	Prediction
MOSAIC-SR	$1.0\allowbreak\cdot x_{1}^{2}\allowbreak\cdot x_{2}^{2}\allowbreak\cdot(x_{1}\allowbreak+x_{2})^{-1}$
PySR	$x_{1}\allowbreak\cdot x_{1}\allowbreak\cdot x_{2}\allowbreak\cdot(1.0000085\allowbreak+\allowbreak-\sin(\cos(x_{2}\allowbreak\cdot x_{2}))\allowbreak\cdot 1.2525511\allowbreak\cdot 10^{-5})\allowbreak\cdot(x_{2}\allowbreak+8.6958323\allowbreak\cdot 10^{-7})\allowbreak\cdot(x_{1}\allowbreak+x_{2})^{-1}$
Operon	$0.011714923195541\allowbreak+0.0004353052779163251485\allowbreak\cdot\exp(-16.278768064237051\allowbreak\cdot x_{1}\allowbreak\cdot x_{2})\allowbreak\cdot\cos(1.651700139045715\allowbreak\cdot x_{1})\allowbreak\cdot x_{2}^{-1}$
GP-GOMEA	$1.0\allowbreak\cdot x_{1}^{2}\allowbreak\cdot x_{2}^{2}\allowbreak\cdot(x_{1}\allowbreak+x_{2})^{-1}$
E2E-SR	$(0.06102565887369925\allowbreak-0.8666989784169188\allowbreak\cdot x_{2})\allowbreak\cdot(5.672968018606769\allowbreak-0.16969952289121198\allowbreak\cdot(-1.101556530062938\allowbreak\cdot x_{1}\allowbreak-1.7747546914056933\allowbreak\cdot x_{2}\allowbreak-1.3923669907018425)^{-1})\allowbreak\cdot(0.08756886213383504\allowbreak\cdot x_{1}\allowbreak-0.366604846256249)\allowbreak\cdot(1.7505584083402312\allowbreak\cdot x_{2}\allowbreak-0.12325958446073971)$
SNIP-SR	$-24535.521299664357\allowbreak\cdot(7.784214446548355\allowbreak\cdot 10^{-5}\allowbreak\cdot x_{1}\allowbreak+1)^{2}\allowbreak+24534.943977652507\allowbreak+4576.952109241969\allowbreak\cdot(10366.057719507926\allowbreak-10583.112972836361\allowbreak\cdot x_{1})^{-1}$

Appendix

Table 42

Method	Prediction
MOSAIC-SR	$\exp(-x_{1}^{2})$
PySR	$-x_{1}\allowbreak+\exp(x_{1}\allowbreak\cdot 2.732575\allowbreak\cdot x_{1}\allowbreak\cdot-2.732575^{-1})\allowbreak+x_{1}\allowbreak\cdot x_{1}\allowbreak\cdot x_{1}^{-1}$
Operon	$0.307795726557961\allowbreak\cdot x_{1}\allowbreak+0.889393270015717$
GP-GOMEA	$0.80009440459554$
E2E-SR	$0.4\allowbreak\cdot\cos(2.1126836069969081\allowbreak\cdot x_{1}\allowbreak-0.03706022875549485)\allowbreak+0.6$
SNIP-SR	$130.29417035635083\allowbreak\cdot(1\allowbreak-0.027990232692061055\allowbreak\cdot x_{1})^{2}\allowbreak-17.620730166228215\allowbreak\cdot(1\allowbreak-0.20671522387448319\allowbreak\cdot x_{1})^{2}\allowbreak-111.70304890826463$

Appendix

Table 43

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May 12, 2026

FePySR: A Neural Feature Extraction Framework for Efficient and Scalable Symbolic Regression

A fundamental challenge in symbolic regression (SR) is efficiently recovering complex mathematical expressions from observational data. Although this problem is NP-hard, many expressions of practical interest decompose naturally into combinations of nonlinear feature modules, concentrating structural complexity into a small number of reusable components. Here, we introduce FePySR, a two-stage framework that reduces the SR search space by extracting valid features prior to equation search. FePySR first employs a heterogeneous neural network to constrain observational data to a set of candidate expressions, then performs structural optimization within this refined expression space using PySR. Across five standard benchmarks, FePySR outperforms state-of-the-art methods by achieving higher equation recovery rates. On a set of 75 highly complex synthesized equations, FePySR recovers 36 equations, while producing substantially smaller mean squared errors on the remaining unrecovered cases, with reduced computation time compared to PySR. FePySR's first stage also maintains consistent performance under varying numbers of selected top features and increasing levels of noise in the observational data. Applied to ordinary differential equations governing biological systems, FePySR successfully identifies governing equations in 24 out of 100 tests where PySR recovers none. Taken together, FePySR is a generalizable framework that can enhance the SR solvers, enabling the efficient and reliable recovery of symbolic expressions across scientific domains.

Zhiming Yu, Wangtao Lu, Xin Lai

School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, China · Faculty of Medicine and Health Technology, Tampere University, Tampere, Finland

Nonlinear System Identification Automated Feature Engineering Symbolic Regression

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Jul 5, 2026

Language models guide symbolic equation discovery by controlling search

Scientific equation discovery must combine broad domain priors with strict numerical testing. Symbolic regression supplies numerical grounding but faces a combinatorial search space, whereas many language-model systems ask the model to propose or select formulas directly. We test a different division of labour. We compare role specifications in which the language model acts as equation author, candidate decider or search controller, alongside end-to-end language-model and purely numerical baselines. In the controller setting we propose here, implemented as LLM-PySR, language models specify variables, operators, transformations and search depth; symbolic regression enumerates and fits expressions; and deterministic metrics govern retention. Across 74 AI-Feynman equations and seven complex formula-recovery tasks, search control achieved the strongest observed balance of accuracy, complexity, stability and cost. On an independent battery dataset, LLM-PySR identified a compact piecewise-linear relation between early voltage-curve displacement and cycle life. The results suggest that language models should shape hypothesis exploration rather than decide which equations survive.

Zikai Xie, Wenmei Li, Man Luo +2

State Key Laboratory of Precision and Intelligent Chemistry, Hefei National Research Center for Physical Sciences at the Microscale, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, China. · Center for Scientific Intelligence Innovation, Hefei, China · School of Chemistry, School of Computer Science, University of Birmingham, Birmingham, UK

Large Language Model-Guided Scientific Discovery Symbolic Regression

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Aug 17, 2026

AutoSR: Automatic Symbolic Regression by Searching Research States

We introduce Automatic Symbolic Regression (AutoSR), a fully automated system that instantiates Research-Space Symbolic Regression by searching persistent scientific investigations rather than isolated equations. Finite, noisy data often yield numerically competitive expressions that imply very different behavior outside the observed regime, making numerical fit and syntactic complexity insufficient measures of scientific credibility. Existing approaches largely focus on improving expressions, yet the search typically retains little beyond the resulting formula and score, losing the scientific record, such as motivations and probes, that inform what to try next. AutoSR preserves this record in a Research State, coupling each candidate equation with the reasoning, computational evidence, and independent review developed along its branch. Proposer--reviewer agents develop these states under progressive-widening Monte Carlo tree search (PW-MCTS), which allocates computation across competing investigations, while the accumulated research record is ultimately synthesized into a final report that explains the leading relation and the basis for its selection. Across nine selected challenges from two benchmark suites, AutoSR recovers algebraically equivalent relations in every case, including three cp3-bench problems that no published system recovers and six structurally diverse LSR-Transform problems. Overall, AutoSR extends symbolic regression from equation-level search toward automated scientific investigation, allowing scientific knowledge and accumulated evidence to shape both what is explored and how the resulting equation is justified.

Kejia Zhang, Youran Sun, Xinyu Ren +2

Department of Mathematics, University of Maryland, College Park, MD, USA · The Chinese University of Hong Kong, Hong Kong SAR, China

Symbolic Regression Autonomous Scientific Discovery Autonomous Research Agents

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