Test-Time Scaling for Scientific Equation Discovery

Haowei Lin; Hubert Lim; Xiangyu Wang; Letian Huang; Di He

cs.CLAug 21, 2026

Test-Time Scaling for Scientific Equation Discovery

Authors: Haowei Lin, Hubert Lim, Xiangyu Wang, Letian Huang, Di He

Organizations: Peking University

Abstract

Test-time scaling (TTS) improves language model reasoning by allocating additional test-time compute, but prior work mainly studies closed-ended tasks such as math and coding. We study TTS for automated equation discovery, an open-ended setting where models search over candidate equations and rely on observed datapoints for feedback. We formulate LLM-driven equation discovery as an iterative search process that unifies Best-of-N, sequential refinement, tree search, and evolution-style methods under a common compute-allocation view. To isolate allocation effects from prompt engineering and other heuristics, we compare minimal parallel controllers under fixed budgets. On LLM-SRBench equation-discovery tasks, we find that search width is the dominant allocation parameter: the best width in our sweep generally increases with the compute budget, while the population--branching split and controller choice matter less. Appropriate width selection also improves wall-clock efficiency by increasing parallelism. These results suggest that, given an informative verifier, controlling exploration and exploitation is central to scaling LLM-based equation discovery.

Figures & tables

Figure 1: Comparison of TTS control flows under the unified formulation. Top row: classical baselines emphasize only one aspect of compute allocation: Best-of-

N

performs purely parallel one-shot sampling (

T=1

n=1

k\ \text{large}

), Self-refine performs purely sequential single-chain refinement (

T=N

n=1

k=1

), and MCTS balances exploration and exploitation through sequential tree expansion but with limited parallelism. Bottom row: hybrid, equation-discovery-oriented methods maintain multiple active candidates and combine branching with pruning. OpenEvolve expands multiple isolated islands in parallel (

n=\#\text{islands}

k=1

); our Parallel Beam Search (PBeam) performs beam-style expansion within each island (

n\geq 1,k\geq 1,g\geq 1

); and Parallel Iterative Expansion (PIE) further relaxes strict beam pruning by expanding from the full historical pool.

Figure 2: Search-width frontiers across domains and backbones. Each curve corresponds to a fixed width

w=nk

(

g=1

), with colors transitioning from dark (small

w

) to bright (large

w

). Stars mark the best attainable point at each budget over both PBeam and PIE. Across the budgets considered, the envelope tends to shift from narrower to wider searches, indicating that width is a key allocation variable.

Figure 3: Wall-clock frontier under different compute-allocation strategies. Each point is the best measured configuration at fixed

(N,w,g,\text{algorithm})

after optimizing over the remaining decomposition. Marker shape indicates the algorithm, color indicates width, and the black curve shows the time–quality Pareto frontier.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Domain	# Tasks	Scientific setting	Inputs	Output	Range summary from task files
Bio	24	Population dynamics	$(t,P)$	$dP/dt$	Each task exposes 4,000 training points. Across tasks, $t$ lies near $[1,54.10]$ , the upper bound of $P$ ranges from $1$ to $996.63$ , and the training output range for $dP/dt$ spans approximately $[-33.35,60.54]$ .
Material	25	Stress–strain relationships	$(\epsilon,T)$	$\sigma$	Each task exposes 4,000 training points. Across tasks, $\epsilon$ lies near $[0,0.54]$ , $T$ lies near $[273,542.99]\,\mathrm{K}$ , and the training output range for $\sigma$ spans approximately $[-606.19,1.36\times 10^{5}]$ .

Appendix

Table 1: Dataset statistics for the two LSR-Synth domains considered in the paper. The input/output signatures are domain-level invariants; the hidden equation and numeric ranges vary across tasks.

Study stage	Configurations
$nk$ assignment	Main width–depth sweep. For each reported model/domain/seed tuple, we fix $g=1$ and enumerate all power-of-two pairs $(n,k)$ satisfying $nk\leq 128$ . This yields 36 PBeam configurations. PIE uses the same grid except for the degenerate $(n,k)=(1,1)$ case and the single-step settings with $nk=128$ , yielding 27 additional configurations. Each run is launched once at maximum budget $128$ , and lower target budgets $N\in\{1,2,4,8,16,32,64,128\}$ are recovered from best-so-far checkpoints by selecting the last logged state with cumulative budget_used $\leq N$ .
G assignment	Grouping ablation, performed only for gpt-oss-20b on Bio and Material. We fix $N=128$ and sweep group counts on the competitive high-width slices identified in the first stage. Width-32 settings are $(n,k)\in\{(32,1),(16,2),(8,4)\}$ and width-16 settings are $(n,k)\in\{(16,1),(8,2)\}$ . For each pair we test every divisor $g$ of $n$ in $\{1,2,4,8,16,32\}$ , producing 24 configurations per algorithm.

Appendix

Table 2: Experimental grid used in the reported study.

ID	Right-hand side	ID	Right-hand side
BPG1	$r\left(1-\frac{P(t)}{K_{0}}\right)P(t)+rP(t)^{0.33}$	BPG13	$\beta P(t)\sin(\omega t)+r\left(1-\frac{P(t)}{K_{0}}\right)P(t)$
BPG2	$rP(t)\exp(-\gamma t)+\frac{rP(t)^{2}}{\alpha P(t)+1}$	BPG14	$r\left(-1+\frac{P(t)}{\alpha}\right)\left(1-\frac{P(t)}{K_{0}}\right)P(t)+rP(t)+\frac{rP(t)}{1+\exp\left(-\alpha(-\beta+P(t))\right)}$
BPG3	$\beta P(t)\sin(\omega t)+rP(t)\exp(-\gamma t)$	BPG15	$r\left(1-\frac{P(t)}{K_{0}}\right)P(t)+r\left(1-\exp(-\gamma P(t))\right)P(t)+rP(t)\exp(-\gamma t)$
BPG4	$r\left(-1+\frac{P(t)}{\alpha}\right)\left(1-\frac{P(t)}{K_{0}}\right)P(t)+r\left(1-\exp(-\gamma P(t))\right)P(t)$	BPG16	$rP(t)^{0.33}+rP(t)\exp(-\gamma t)$
BPG5	$r\left(1-\frac{P(t)}{K_{0}}\right)P(t)+\frac{rP(t)}{1+\exp\left(-\alpha(-\beta+P(t))\right)}$	BPG17	$r\left(-1+\frac{P(t)}{\alpha}\right)\left(1-\frac{P(t)}{K_{0}}\right)P(t)+rP(t)^{0.33}+rP(t)$
BPG6	$r\left(1-\frac{P(t)}{K_{0}}\right)P(t)+\frac{rP(t)^{2}}{\alpha P(t)+1}$	BPG18	$r\left(-1+\frac{P(t)}{\alpha}\right)\left(1-\frac{P(t)}{K_{0}}\right)P(t)+rP(t)^{0.33}$

Appendix

Table 3: Biology-domain LSR-Synth equations used in this paper. Each entry reports the right-hand side of

\frac{dP}{dt}=f(t,P)

ID	Right-hand side	ID	Right-hand side
MatSci1	$E_{0}\epsilon\left(-\alpha_{T}(T-T_{0})+1\right)-\beta(T-T_{0})+\epsilon^{M}\eta(T-T_{0})$	MatSci14	$-\beta(T-T_{0})+\epsilon\eta\exp\left(-(T-T_{0})^{2}\right)$
MatSci2	$H\epsilon^{3}+K\epsilon^{N}\exp\left(-\frac{Q}{RT}\right)+\epsilon\eta\sin(T-T_{0})$	MatSci15	$-\beta(T-T_{0})+\epsilon^{M}\eta(T-T_{0})$
MatSci3	$H\epsilon^{3}+\eta(T-T_{0})\exp(-\epsilon)$	MatSci16	$E_{0}\epsilon\left(-\alpha_{T}(T-T_{0})+1\right)+\epsilon\eta\exp\left(-(T-T_{0})^{2}\right)$
MatSci4	$H\epsilon^{3}+K\epsilon^{N}\exp\left(-\frac{Q}{RT}\right)+\epsilon^{3}\eta(T-T_{0})$	MatSci17	$E_{0}\epsilon^{2}+\epsilon\eta(T-T_{0})^{2}$
MatSci5	$E_{0}\epsilon^{2}+\eta(T-T_{0})\log(\epsilon+1)$	MatSci18	$E_{0}\epsilon\left(-\alpha_{T}(T-T_{0})+1\right)-\beta(T-T_{0})+\eta(T-T_{0})\log(\epsilon+1)$
MatSci6	$E_{0}\epsilon\left(-\alpha_{T}(T-T_{0})+1\right)+K\epsilon^{N}\exp\left(-\frac{Q}{RT}\right)+\epsilon^{M}\eta(T-T_{0})$	MatSci19	$H\epsilon^{3}+\eta(T-T_{0})\sin(\epsilon)$

Appendix

Table 4: Material-domain LSR-Synth equations used in this paper. Each entry reports the corresponding stress law

\sigma(\epsilon,T)=g(\epsilon,T)

	Bio			Material
Budget $N$	Algorithm	$(n,k,T)$	$w^{\star}$	Algorithm	$(n,k,T)$	$w^{\star}$
1	PBeam	$(1,1,1)$	1	PBeam	$(1,1,1)$	1
2	PBeam	$(1,1,2)$	1	PBeam	$(1,1,2)$	1
4	PBeam	$(1,2,2)$	2	PBeam	$(2,1,2)$	2
8	PIE	$(1,2,4)$	2	PIE	$(2,2,2)$	4
16	PIE	$(1,2,8)$	2	PBeam	$(1,4,4)$	4

Appendix

Table 5: Exact optimal width on gpt-oss-20b without grouping. For each pair

(N,w)

, we first optimize over all decompositions

(n,k)

inside PBeam and inside PIE, and then choose the better algorithm. The table therefore reports which algorithm attains the envelope at each budget rather than a globally preferred control flow.

Domain	Algorithm	Regime	Best $(n,k)$	Train $\operatorname{Acc}_{0.1}$	Rel. gap
Bio	PBeam	Population-heavy	$(16,2)$	0.9816	0.28%
Bio	PBeam	Balanced	$(4,8)$	0.9844	0.00%
Bio	PBeam	Branching-heavy	$(1,32)$	0.9801	0.44%
Bio	PIE	Population-heavy	$(16,2)$	0.9804	0.00%
Bio	PIE	Balanced	$(8,4)$	0.9612	1.95%
Bio	PIE	Branching-heavy	$(2,16)$	0.9794	0.10%

Appendix

Table 6: Best

(n,k)

decomposition within each regime at fixed budget

N=128

, width

w=32

, and no grouping (

g=1

). For each domain and algorithm, we partition the six admissible decompositions of

w=32

into three regimes and report the best

(n,k)

and its train

\operatorname{Acc}_{0.1}

within each regime. Relative gap is computed against the best regime under the same domain and algorithm.

Figure 4: PIE

-

PBeam after optimizing

(n,k)

inside each algorithm. The vertical axis shows train

\operatorname{Acc}_{0.1}

differences in percentage points. The effect size is much smaller than the effect of choosing the right width.

Figure 5: Matched comparison among PBeam, PIE, and OpenEvolve.

Figure 6: Effect of group size on performance and wall-clock time. For each domain and width, every point is the best measured configuration at fixed

(w,g)

and

N=128

, compared against the best ungrouped configuration with the same width. Moderate grouping can improve throughput with modest quality loss, but excessively large

g

eventually collapses the per-group width and hurts performance.

Figure 7: Train vs. test performance of gpt-oss-20b. Mean accuracy (

\operatorname{Acc}_{0.1}

) on the training and test sets for the Bio (left) and Material (right) domains across varying compute budgets

N

Figure 8: Auxiliary cross-model comparison on Bio with Qwen3-30B-A3B . Solid lines show the exact best-performing width in the sweep. The same qualitative width–depth trade-off appears across models, although the exact mid-budget operating point remains backbone-dependent.

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