OptScientist: Automated Discovery of Optimization Laws

Ma ZhenYao; Yue Liang; Jian-Qiao Zhu

cs.AIOct 20, 2026

OptScientist: Automated Discovery of Optimization Laws

Authors: Ma ZhenYao, Yue Liang, Jian-Qiao Zhu

Organizations: Plokerm AI Lab · Xiamen University · University of Tübingen · The University of Hong Kong

Abstract

Many scientific phenomena are best understood as outcomes of underlying optimization problems, yet automated scientific discovery has largely focused on recovering equations that describe observed input-output patterns. We introduce OptScientist, a novel automated framework for discovering compact symbolic optimization laws directly from data. OptScientist comprises (i) a model that infers symbolic optimization problems by first learning to represent data as an optimization problem and then extracting its symbolic form, and (ii) an LLM agentic system embedding the model in a discovery process that integrates theoretical priors from scientific literature, discovers candidate laws, abstracts their underlying mechanisms, and suggests new data to collect. We first evaluate OptScientist on OptFeynman, a novel benchmark of 99 physics-derived optimization problems, where OptScientist recovers ground-truth laws for over 80% of the problems and remains robust to noisy contextual information, while substantially outperforming machine-learning, symbolic, and LLM-based model-discovery baselines. On human risky choice and central-bank monetary policy, OptScientist discovers compact symbolic optimization laws that achieve held-out predictive performance competitive with leading baselines, while using few parameters and substantially fewer tokens than LLM baselines. Overall, OptScientist extends automated scientific discovery to uncovering optimization laws that generate observed phenomena, enabling automated theory building across the physical, biological, behavioral, and social sciences.

Figures & tables

Figure 1: The OptScientist Model. a, The OptScientist Model consists of an Optimization Learner and an Optimization Discoverer. The Optimization Learner learns to represent observational data as an optimization problem using a hierarchical network with an optimization representation bottleneck. The Optimization Discoverer then distills this learned optimization representation into a compact symbolic optimization problem. b, The OptScientist Model represents the data manifold through a latent optimization network. Raw observations are first compressed and represented by hierarchical optimization blocks. The learned representations are then expressed as a single optimization problem through the optimization representation bottleneck, from which a compact symbolic law is extracted. c, The model follows the same logic in concrete discovery settings. Noisy real-world observations are first transformed into denoised representations, and symbolic regression is then applied to the learned optimization representation to recover symbolic laws.

Figure 2: OptScientist Agent as an iterative research framework. The framework embeds the OptScientist Model in a scientific-discovery loop.

Figure 3: Recovering ground-truth optimization laws on OptFeynman. a, OptFeynman converts analytic physics equations into latent optimization problems and generates observations by Gibbs sampling from the induced landscapes. b,c, For discrete decision variables, OptScientist shows the strongest out-of-sample ability to recover the true optimization law on unseen test contexts, outperforming baselines. It achieves perfect exact-order recovery on $88/99$ tasks, indicating that the recovered symbolic laws induce preference orders approximately equivalent to the ground-truth laws. d,e, For continuous decision variables, the advantage persists on unseen test contexts across increasingly dense candidate sets. OptScientist perfectly recovers the exact order on $78/99$ tasks at $K=10$ , $K=50$ and $K=100$ , showing robust discovery of the underlying optimization law beyond discrete-choice settings. Error bars denote standard errors across datasets where shown.

Figure 4: Robust recovery of optimization laws under noisy context variables. Gaussian noise is added to the training context variables, while validation and test contexts remain clean. a, For discrete decision variables, OptScientist remains the strongest method for out-of-sample recovery of the true optimization law, achieving perfect exact-order recovery on

92/99

tasks and outperforming machine-learning, symbolic-regression, optimization-structured, and LLM-based automated model-discovery baselines. b,c, For continuous decision variables, OptScientist retains high recovery performance on unseen test contexts across

K=10

K=50

and

K=100

, perfectly recovering the exact order on

76/99

tasks at each candidate-set size and continuing to outperform the evaluated baselines. These results show that the model can recover approximately equivalent ground-truth optimization laws even when the training observations are corrupted by context noise. Error bars denote standard errors across datasets where shown.

Decision variable	Effective ground truth	Effective recovered expression
$y_{3}$	$\displaystyle\max_{y_{3}}\;x_{3}y_{3}$	$\displaystyle\max_{y_{3}}\;x_{3}y_{3}$
$w$	$\displaystyle\max_{w}\;w^{2}$	$\displaystyle\max_{w}\;w^{2}$
$E_{f}$	$\displaystyle\max_{E_{f}}\;E_{f}\cos\theta$	$\displaystyle\max_{E_{f}}\;E_{f}\cos\theta$
$G$	$\displaystyle\max_{G}\;G\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right)$	$\displaystyle\max_{G}\;\frac{G(r_{1}-r_{2})}{r_{1}r_{2}}$
$d$	$\displaystyle\max_{d}\;\frac{T_{2}-T_{1}}{d}$	$\displaystyle\max_{d}\;\frac{T_{2}-T_{1}}{d}$
$r$	$\displaystyle\max_{r}\;\frac{1}{r^{2}}$	$\displaystyle\max_{r}\;\frac{1}{r^{2}}$

Table 1: Examples of recovered OptFeynman laws. Examples show that OptScientist perfectly recovers effective ground-truth optimization expressions across diverse physical structures.

Figure 5: A compact law of human decision-making under risk and ambiguity. a, A preliminary analysis of the CPC18 repeated-choice task shows that decision-history variables are particularly informative relative to lottery, demographic and framing variables, consistent with the sequential structure of the discovered optimization law. b, OptScientist recovers this sequential structure as a compact symbolic law: people tend to follow immediate reward, resist switching from a previous choice and remain sensitive to forgone outcomes. c, Across behavioural baselines, machine-learning models and OptScientist variants, the recovered law achieves the near-best predictive performance with a near-minimal number of parameters. The result shows that human choice can be compressed into a simple optimization principle rather than explained by model scale alone. Error bars denote the standard error of the mean across participant-level metrics.

Figure 6: Discovery of a central-bank policy law. a, OptScientist recovers a compact symbolic rule for quarterly monetary-policy decisions. The five-class policy stance, ranging from large easing to large tightening, is explained as tracking a latent policy target driven by macroeconomic strength, financial fragility, inherited tightness and policy momentum. b, On a future-period out-of-time test set, OptScientist attains the best policy-stance and policy-direction accuracy while using a near-minimal parameter count. Error bars denote bootstrap standard errors from

10{,}000

quarter-level resamples with replacement.

Figure 7: Token-efficient automated model discovery. We further compare the token efficiency of OptScientist with three LLM-based automated model-discovery methods on the two real-world discovery tasks: LLM-only discovery, GeCCo (code-assisted LLM), and MDA (Bayesian experiment design). All LLM components use DeepSeek-V4-Flash. Left, On the central-bank monetary-policy task (Sec. 4.3 ), OptScientist attains higher held-out policy-stance accuracy with substantially lower output-token consumption. Right, On human decisions under risk and ambiguity (Sec. 4.2 ), OptScientist reaches competitive held-out predictive accuracy with substantially fewer output tokens. Across both domains, OptScientist achieves a favourable accuracy–token-efficiency trade-off.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

\mathcal{L}=-\frac{1}{|\mathcal{B}|}\sum_{i\in\mathcal{B}}\log\frac{\exp(S_{\theta}(x_{i},y_{i})/\tau)}{\sum_{y\in\mathcal{A}_{i}}\exp(S_{\theta}(x_{i},y)/\tau)}

Appendix

Algorithm 1 OptScientist Model

OptLearner architecture
Hyperparameter	Discrete	Continuous
Number of BL layers	2	2
BL layer widths	$[128,1]$	$[128,1]$
Hidden-layer $(U,C,T)$ branches	$(1,1,0)$	$(1,1,0)$
Default final-layer $(U,C,T)$ branches	$(1,0,0)$	$(1,0,0)$
$U$ activation	Identity	Identity

Appendix

Table 2: Default OptScientist Model hyperparameters. The discrete and continuous implementations use the same OptLearner architecture and training procedure, with different candidate construction for the learning objective. Symbolic operators are task-dependent; the table reports the default base language, and additional binary or unary operators can be introduced when required by the discovery problem. The default final block contains a single objective component

U

, while task-specific configurations can additionally activate

C

and

T

components.

Figure 8: OptLearner learning curves on OptFeynman. Training and validation losses for continuous and discrete OptFeynman tasks, with and without noisy context variables. Loss is the OptLearner objective defined in the main text:

L_{\mathrm{cont}}

for continuous outcomes and

L_{\mathrm{disc}}

for discrete outcomes. For each task, losses are normalized by their values at the first epoch, and training progress is normalized from the first epoch to the epoch selected by validation-based early stopping. Curves show the median across tasks and shaded regions show the interquartile range.

Decision type	Representation	Candidate set	Clean	Noisy-context
Discrete	OptScientist law	$K=5$	$88/99\;(88.89\%)$	$92/99\;(92.93\%)$
Discrete	OptLearner score	$K=5$	$31/99\;(31.31\%)$	$27/99\;(27.27\%)$
Continuous	OptScientist law	$K=10$	$78/99\;(78.79\%)$	$76/99\;(76.77\%)$
Continuous	OptLearner score	$K=10$	$30/99\;(30.30\%)$	$30/99\;(30.30\%)$
Continuous	OptScientist law	$K=50$	$78/99\;(78.79\%)$	$76/99\;(76.77\%)$
Continuous	OptLearner score	$K=50$	$20/99\;(20.20\%)$	$23/99\;(23.23\%)$

Appendix

Table 3: Perfect exact-order recovery on OptFeynman. Entries report the number and percentage of the 99 tasks with exact-order accuracy equal to one. OptScientist law denotes the final symbolic optimization law produced by OptDiscover.

Condition	$K$	Representation	Exact order	Median [Q1,Q3]	Pairwise
Clean	10	OptScientist law	$0.900\pm 0.027$	$1.000\,[1.000,1.000]$	$0.963\pm 0.015$
Clean	10	OptLearner score	$0.825\pm 0.026$	$0.940\,[0.800,1.000]$	$0.960\pm 0.007$
Clean	50	OptScientist law	$0.871\pm 0.032$	$1.000\,[1.000,1.000]$	$0.963\pm 0.015$
Clean	50	OptLearner score	$0.743\pm 0.035$	$0.900\,[0.695,0.990]$	$0.960\pm 0.008$
Clean	100	OptScientist law	$0.866\pm 0.033$	$1.000\,[1.000,1.000]$	$0.963\pm 0.015$
Clean	100	OptLearner score	$0.719\pm 0.035$	$0.880\,[0.665,0.970]$	$0.960\pm 0.008$

Appendix

Table 4: Preference recovery for continuous OptFeynman problems. Values are mean

\pm

SE across 99 OptFeynman problems. Median and interquartile range refer to exact-order accuracy. Higher values indicate better recovery.

Condition	Representation	KL	Median [Q1,Q3]
Clean	OptScientist law	$0.0461\pm 0.0111$	$0.0080\,[0.0013,0.0358]$
Clean	OptLearner score	$0.0078\pm 0.0032$	$0.0015\,[0.0010,0.0030]$
Noisy-context	OptScientist law	$0.0460\pm 0.0163$	$0.0071\,[0.0012,0.0382]$
Noisy-context	OptLearner score	$0.0082\pm 0.0036$	$0.0014\,[0.0009,0.0031]$

Appendix

Table 5: Distribution recovery for continuous OptFeynman problems. KL divergence is computed between the ground-truth and recovered Gibbs distributions on the dense test grid. Values are mean

\pm

SE across 99 OptFeynman problems. Lower values indicate closer distribution recovery.

Condition	Representation	Exact order	Median [Q1,Q3]	Pairwise	Exact $=1$
Clean	OptScientist law	$0.971\pm 0.014$	$1.000\,[1.000,1.000]$	$0.984\pm 0.011$	$88/99$
Clean	OptLearner score	$0.805\pm 0.025$	$0.920\,[0.675,1.000]$	$0.947\pm 0.009$	$31/99$
Noisy-context	OptScientist law	$0.974\pm 0.015$	$1.000\,[1.000,1.000]$	$0.984\pm 0.011$	$92/99$
Noisy-context	OptLearner score	$0.805\pm 0.024$	$0.890\,[0.705,1.000]$	$0.948\pm 0.009$	$27/99$

Appendix

Table 6: Preference recovery for discrete OptFeynman problems. Values are mean

\pm

SE across 99 problems. Median and interquartile range refer to exact-order accuracy.

Condition	Representation	Prob. MAE	MAE median [Q1,Q3]	KL	KL median [Q1,Q3]
Clean	OptScientist law	$0.0182\pm 0.0025$	$0.0115\,[0.0061,0.0201]$	$0.0217\pm 0.0092$	$0.0036\,[0.0008,0.0109]$
Clean	OptLearner score	$0.0161\pm 0.0032$	$0.0112\,[0.0083,0.0154]$	$0.1932\pm 0.1873$	$0.0032\,[0.0016,0.0059]$
Noisy-context	OptScientist law	$0.0208\pm 0.0036$	$0.0117\,[0.0062,0.0238]$	$0.2005\pm 0.1852$	$0.0036\,[0.0010,0.0132]$
Noisy-context	OptLearner score	$0.0161\pm 0.0032$	$0.0110\,[0.0080,0.0146]$	$0.2025\pm 0.1964$	$0.0033\,[0.0016,0.0052]$

Appendix

Table 7: Distribution recovery for discrete OptFeynman problems. The ground-truth Gibbs distribution over five candidate actions is compared with the distribution induced by the recovered score. Probability MAE is the mean absolute error between ground-truth and recovered Gibbs probabilities. Values are mean

\pm

SE across 99 problems. Lower values indicate closer recovery.

Decision variable	Effective ground truth	Effective recovered expression
$y_{3}$	$\displaystyle\max_{y_{3}}\;x_{3}y_{3}$	$\displaystyle\max_{y_{3}}\;x_{3}y_{3}$
$w$	$\displaystyle\max_{w}\;w^{2}$	$\displaystyle\max_{w}\;w^{2}$
$\chi$	$\displaystyle\max_{\chi}\;B\chi$	$\displaystyle\max_{\chi}\;B\chi$
$E_{f}$	$\displaystyle\max_{E_{f}}\;E_{f}\cos\theta$	$\displaystyle\max_{E_{f}}\;E_{f}\cos\theta$
$\theta$	$\displaystyle\max_{\theta}\;\sin\theta$	$\displaystyle\max_{\theta}\;\sin\theta$
$G$	$\displaystyle\max_{G}\;G\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right)$	$\displaystyle\max_{G}\;\frac{G(r_{1}-r_{2})}{r_{1}r_{2}}$

Appendix

Table 8: Examples of recovered OptFeynman laws. Ground-truth and recovered laws are shown in optimization form after removing numerical constants and scale factors that do not affect the optimizer.

Decision type	Condition	Initial	Final	$\Delta$	Perfect recovery
Continuous	Clean, $K=10$	$0.890$	$0.900$	$+0.010$	$77\rightarrow 78$
Continuous	Clean, $K=50$	$0.860$	$0.871$	$+0.011$	$72\rightarrow 78$
Continuous	Clean, $K=100$	$0.852$	$0.866$	$+0.013$	$67\rightarrow 78$
Continuous	Noisy-context, $K=10$	$0.902$	$0.902$	$-0.001$	$76\rightarrow 76$
Continuous	Noisy-context, $K=50$	$0.872$	$0.874$	$+0.002$	$75\rightarrow 76$
Continuous	Noisy-context, $K=100$	$0.864$	$0.869$	$+0.005$	$70\rightarrow 76$

Appendix

Table 9: Exact-order recovery before and after OptDiscover complexity control. Mean accuracy is computed across matched OptFeynman problems. Initial denotes the recovered symbolic expression before expression-level reduction, and Final denotes the compact symbolic expression after it. The final column reports the number of problems with perfect exact-order recovery before and after this reduction.

Condition	Method	$n$	Exact order	Exact $=1$	Pairwise	KL
Clean	GBDT	99	0.327	0.0%	0.844	0.1679
Clean	LR	99	0.716	17.2%	0.919	0.0859
Clean	MLP	99	0.420	6.1%	0.859	0.1580
Clean	RF	99	0.206	0.0%	0.794	0.5992
Clean	XGBoost	99	0.337	0.0%	0.847	0.0833
Clean	SR	99	0.076	0.0%	0.743	0.8855

Appendix

Table 10: Held-out baseline recovery for discrete OptFeynman problems. Exact-order and pairwise preference accuracy are evaluated on unseen test contexts. Exact

=1

reports the percentage of tasks with perfect exact-order recovery. KL divergence compares the ground-truth and recovered Gibbs distributions over the five candidate actions; lower values indicate closer distribution recovery. For discrete decisions, a standard multiclass MLP can be written in the same finite-action softmax form as a conditional Neural EBM, so we use the multiclass MLP directly rather than introducing a separate Neural EBM baseline.

Condition	Method	$n$	$\text{Exact order}_{10/50/100}$	$\text{Exact}=1~(\%)_{10/50/100}$	$\text{Pairwise}_{10/50/100}$	KL
Clean	GBDT	99	$0.063/0.003/0.002$	$0.0/0.0/0.0$	$0.704/0.704/0.704$	0.6362
Clean	Neural EBM	99	$0.482/0.314/0.294$	$12.1/10.1/10.1$	$0.888/0.887/0.888$	0.0098
Clean	MLP	99	$0.061/0.003/0.001$	$0.0/0.0/0.0$	$0.703/0.703/0.703$	0.6349
Clean	RF	99	$0.064/0.003/0.001$	$0.0/0.0/0.0$	$0.703/0.702/0.703$	0.6408
Clean	XGBoost	99	$0.061/0.003/0.001$	$0.0/0.0/0.0$	$0.704/0.704/0.704$	0.6399
Clean	SR	99	$0.060/0.002/0.001$	$0.0/0.0/0.0$	$0.710/0.710/0.711$	0.6414

Appendix

Table 11: Held-out baseline recovery for continuous OptFeynman problems. Within the exact-order, perfect-recovery and pairwise columns, entries correspond to

K=10/50/100

, respectively. Exact

=1

reports the percentage of tasks with perfect exact-order recovery. KL divergence compares the ground-truth and recovered Gibbs distributions on the dense held-out test grid; lower values indicate closer distribution recovery.

Variant	Parameters	Accuracy $\uparrow$	AUC $\uparrow$	Prob. MSE $\downarrow$	NLL $\downarrow$	KL $\downarrow$	ECE $\downarrow$
Direct OptScientist	5	$0.829\pm 0.004$	$0.863\pm 0.004$	$0.149\pm 0.002$	$0.476\pm 0.004$	$0.011\pm 0.001$	$0.172\pm 0.002$
AutoF OptScientist	7	$0.798\pm 0.004$	$0.865\pm 0.004$	$0.137\pm 0.002$	$0.422\pm 0.006$	$0.006\pm 0.000$	$0.116\pm 0.001$

Appendix

Table 12: Predictive performance and complexity on CPC18. Metrics are reported as participant-level mean

\pm

standard error across test participants. Parameter count denotes the number of distinct numerical constants in the recovered symbolic law.

Model	Params.	MSE $\downarrow$	MAE $\downarrow$	$R^{2}$ $\uparrow$	Exact acc. $\uparrow$	Direction acc. $\uparrow$
MLP	5,057	$1.6093\pm 0.6026$	$0.9453\pm 0.2250$	$-0.1554\pm 0.5005$	$0.3571\pm 0.1268$	$0.4286\pm 0.1313$
Random forest	134,000	$1.0960\pm 0.3639$	$0.8340\pm 0.1693$	$0.2132\pm 0.1663$	$0.3571\pm 0.1277$	$0.5714\pm 0.1323$
XGBoost	6,600	$0.8501\pm 0.2630$	$0.7616\pm 0.1387$	$0.3897\pm 0.1634$	$0.3571\pm 0.1284$	$0.6429\pm 0.1270$
Neural EBM	5,121	$1.0276\pm 0.4118$	$0.6343\pm 0.2132$	$0.2622\pm 0.3688$	$0.5714\pm 0.1327$	$0.7143\pm 0.1213$
GBDT	6,600	$0.7595\pm 0.2643$	$0.6884\pm 0.1430$	$0.4547\pm 0.1301$	$0.5714\pm 0.1314$	$0.7857\pm 0.1094$
Symbolic regression	6	$0.6788\pm 0.3428$	$\mathbf{0.5636\pm 0.1616}$	$0.5126\pm 0.2009$	$0.5714\pm 0.1322$	$0.7143\pm 0.1204$

Appendix

Table 13: Predictive performance on central-bank monetary-policy decisions. Exact accuracy is computed after mapping continuous predictions to the five policy classes. Direction accuracy compares easing, no-change and tightening directions. Values are mean

\pm

bootstrap standard error over 10,000 quarter-level resamples. OptScientist-C denotes OptScientist with constraints.

Model	Max tokens	Output tokens	Held-out acc. $\uparrow$	Acc./token $(\times 10^{-5})$ $\uparrow$
LLM-only	8,192	17,981	83.354%	4.636
	12,000	15,771	82.956%	5.260
	16,384	16,048	83.173%	5.183
	24,000	16,124	83.259%	5.164
GeCCo ( Rmus et al., 2026 )	8,192	35,947	83.777%	2.331
	12,000	37,317	83.487%	2.237

Appendix

Table 14: Token-efficient model discovery on the CPC18 dataset. Held-out predictive accuracy and cumulative LLM output-token consumption are reported for each discovery setting. The maximum-token column gives the generation limit per LLM call. OptScientist achieves held-out accuracy near the apparent CPC18 predictive ceiling with substantially lower output-token consumption.

Model	Max tokens	Output tokens	Held-out acc. $\uparrow$	Acc./token $(\times 10^{-5})$ $\uparrow$
LLM-only	8,192	18,381	44.318%	2.411
	12,000	16,691	49.432%	2.962
	16,384	17,361	47.727%	2.749
	24,000	18,142	40.341%	2.224
GeCCo ( Rmus et al., 2026 )	8,192	23,188	64.205%	2.769
	12,000	26,350	56.818%	2.156

Appendix

Table 15: Token-efficient model discovery on central-bank monetary-policy decisions. Held-out policy-stance accuracy and cumulative LLM output-token consumption are reported for each discovery setting. OptScientist achieves the highest held-out point estimate while using substantially fewer output tokens than the LLM-based discovery baselines.

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