How AI Agents Discover Scientific Equations: From Hydrotope Rediscovery to New Water-Wave Amplitudes

Zihan Zhou; Digvijay Wadekar; Matias Zaldarriaga

hep-thSep 30, 2026

How AI Agents Discover Scientific Equations: From Hydrotope Rediscovery to New Water-Wave Amplitudes

Authors: Zihan Zhou, Digvijay Wadekar, Matias Zaldarriaga

Organizations: Department of Physics, Princeton University, Princeton, NJ 08540, USA · Center for Gravitational Physics, University of Texas at Austin, Austin, TX 78712, USA · School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

Abstract

We study how AI agents discover and validate scientific formulas using a controlled case study of the hydrotope, a recently discovered geometric formula that combines the different polynomial pieces of nonlinear surface-wave scattering into one global expression. This problem is deceptively difficult: simple formulas can hold within individual frequency regions, but the global result must identify their boundaries and combine exponentially many potentially active terms. We reconstruct how the formula was originally discovered through human--agent collaboration and analyze 18 single-prompt rediscovery runs under no hint and two forms of human guidance: a false hint representing an incorrect prior and a true hint representing domain-informed insight. Only four recover the formula across all kinematic chambers (i.e., regions in which a single polynomial form applies), while most unsuccessful runs find correct chamber polynomials but fail to combine them or test their full domain. Conventional and LLM-assisted symbolic regression and standard machine-learning regressors likewise fail to recover the global formula in our experiments. Guided by these failure modes, we test a PI $+$ two-student workflow in which a coordinating lead agent assigns complementary analytic and numerical tasks to two research agents and independently evaluates their results. The PI $+$ two-student team successfully rediscovers the complete hydrotope formula, while the same workflow applied to the harder three negative wavenumber problem discovers a new independent verified analytic expression for the six-point amplitude $A_6$ .

Figures & tables

Figure 1: This paper asks how AI agents can infer an analytic scattering formula from numerical values provided at different input frequencies. (a) Berends–Giele (BG) recursion numerically calculates the exact answer at any allowed point but does not reveal the analytic formula by itself. A simple formula can work in one frequency chamber and fail in another, so the task is to find one expression valid in every chamber and for an arbitrary number of waves. (b) Fifteen of 18 single-agent runs find a correct formula in at least one chamber, but only four combine the chamber formulas and test the result in new chambers. (c) Two students try different approaches and write their formulas, tests, and failures in a shared record. A PI combines their work and checks it on new examples. Neither of six single-agent instances finds the complete all-

n

hydrotope formula, whereas both PI

+

student agentic teams do. The same team also produces a new six-point three-minus formula that passes 140 exact tests in 58 chambers.

Panel of Figure 2 — Figure 2: Geometric interpretation of the two-minus hydrotope, the known formula for the sector in which two waves have negative wavenumber. A chamber-dependent scattering amplitude can be represented by a slice of a box whose edge lengths are set by the wave frequencies. Crossing a chamber boundary switches a term $(x)_{+}=\max(x,0)$ in Eq. 3 on or off, so the sum adds the active contributions and subtracts their overlaps. At $n=5$ , a plane cuts a three-dimensional box into four allowed polygon types, with the conservation laws excluding the central hexagon; at $n=6$ , the analogous cut is in four dimensions and gives different three-dimensional solids in different chambers. One formula built from $\max(x,0)$ therefore describes every chamber-dependent cross-section.

package	equation guidance	recommended tests
false hint	one ratio of polynomials for all frequencies; no chamber split	comparable frequencies; avoid separated scales and chamber boundaries
true hint	different polynomials in different chambers, with fixed scaling	find the chamber boundaries and test on both sides
no hint	no proposed equation class	test $n=4,5,6,7$ , including widely separated frequency scales

Table 1: The repeated runs vary both the suggested type of equation and the recommended tests. Appendix A reproduces the complete prompt descriptions.

Figure 3: As a benchmark to compare the performance of the agentic method, we test symbolic regression and classical ML methods (see section 3.2 ) to see if they can learn the relationship predicted by the exact (i.e., zero error) five-point hydrotope formula discovered by the multi-agent system (see Eq. ( 5 )). We use the same training dataset used in the agentic runs: wave frequencies as input and corresponding output amplitude from the recursion (BG) relation. Bars show median held-out mean absolute error (MAE), open markers show individual runs, and the triangle marks an off-scale run retained in the benchmark table. Operon has the lowest fitted median MAE (1.36), but no conventional, LLM-assisted, or prediction method recovers exact formula.

condition	complete	partial sum	one chamber	hint rejected	incorrect
false hint	0	1	3	1	1
true hint	4	1	0	0	1
no hint	0	2	4	0	0
total	4	4	7	1	2

Table 2: Each of the three prompts is tested with six agent configurations. “Complete” means correct for every chamber and every

n

. “Partial sum” means that the agent finds the alternating-sum structure but misses a term, an on/off condition, or a test case. “One chamber” means that the final all-

n

formula applies only in one chamber. All four complete results use the true hint that says a different polynomial applies in each chamber.

Figure 4: The rows show eight key steps in order of completeness of the solution. These range from sampling exact values from the BG recursion (K1) to testing and verifying the output analytic formula holds in new chambers (K8). Dark, light, and pale cells mean complete, partial, and not reached. The result column uses C for a complete formula, P for a partial sum, O for a one-chamber formula, R for a rejected false hint, and I for an incorrect result. The final column gives the mean absolute error (MAE) of the run’s

n=8

formula on a separate set of 250 test points; a dash means that the final answer contains no formula that can be evaluated at

n=8

. The largest difference among runs appears at K6–K7, where they must find every boundary and combine the chamber terms.

Figure 5: Four representative runs without human hints all recover the one-term formula in the principal chamber, where no chamber-boundary correction has yet switched on, and find counterexamples outside that chamber. Codex 5.5 and Fugu then produce short but incomplete subset sums. The Claude runs also attempt one formula for all chambers: Claude max’s sorting-based formula misses six of 233 tests, while Claude ultra cannot combine its separate chamber expressions. Both then report only the principal-chamber formula. The bottom expression omits the common factor

i\,2^{n-1}g^{3-n}\omega_{1}\omega_{2}

. Thus, these runs diagnose chamber dependence but fail to synthesize and verify one globally valid formula.

Figure 6: Workflow of our PI+two student (S1, S2) architectures with different base models. The two students construct formulas, while a PI assigns work and checks the result independently. The top row shows the six no-hint single-agent results for comparison, and each box records one team round. The Opus team proposes the complete formula in round 1 and verifies it in round 2; the Codex 5.5 team uses its first failure to divide round 2 between the special

n=4

limit and the complete formula, then the PI independently tests both results and accepts them in round 3. Thus both coordinated teams succeed where none of the six no-hint single agents does, although the runs are not cost-matched and are too few to measure success rates. A written record, two simultaneous searches, and separate final checks help the teams complete the formula.

Figure 7: After rediscovering the known hydrotope formula for two negative wavenumbers, the agents tackle the harder sector with three negative wavenumbers and propose a new analytic expression for the six-point amplitude

A_{6}

intended to cover every chamber (see Eq. ( 18 )). In this eight-round run, rounds 1–4 determine the formula’s ratio structure: a common frequency-dependent denominator and a polynomial numerator carrying the remaining frequency dependence. A failed test in round 5 reveals that the numerator needs corrections involving pairs of chamber boundaries. Rounds 6–7 add those terms, and independently written BG recursion code finds exact agreement on 140 tests in 58 chambers. Round 8 studies

n=7

, whose explicit numerator remains unknown. The failed cross-chamber test drives the correction that completes the supported

A_{6}

expression, but the finite tests do not provide an analytic proof and the numerator for

n\geq 7

remains open.

Figure 8: AI agents often find formulas that work in one frequency chamber but fail to combine them into a global expression. We compare four architectures on the harder discovery problem introduced in Figure 7 : deriving an explicit formula for the new six-point amplitude

A_{6}

in the three-negative-wavenumber sector and testing it in unseen chambers. The single-agent run returns a tree expansion that hides the chamber changes, while the PI

+

one-student team finds separate chamber formulas that fail in a new chamber. Both PI

+

two-student setups, one with an additional verifier, complete and independently check the global expression. Because each architecture was run once and the runs are not cost-matched, these outcomes identify failure modes rather than success rates. In these runs, only teams with two complementary student roles carry both formula construction and cross-chamber testing through to a complete

A_{6}

formula.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 9: This figure repeats the comparison in Figure 3 for the

n=7

case. The formula increases from eight

\max(x,0)

terms at

n=5

to 32 such terms. Bars show median held-out MAE from raw frequencies and open markers show runs; gplearn has the lowest fitted median, 373.49, but slightly negative held-out

R^{2}

and does not recover the target equation. The research team’s exact formula remains at zero, while every fitted method misses terms that switch on in some chambers. A few unusually large values in the fixed split also make absolute errors larger than at

n=5

. Increasing the number of terms that can switch on or off makes exact recovery more difficult under the tested budgets.

Figure 10: Additional methods test whether searches that allow a wider range of equations, including separate equations for different regions, recover the five-point formula. Bars show median held-out MAE and open markers show individual runs for alternative symbolic-regression systems, combinations of region-finding and equation search, archived LLM-assisted variants, SciExplorer, and additional ML methods. PS-Tree improves approximate prediction by learning seven regions with separate symbolic expressions, but it does not recover the compact hydrotope formula; the region-first Operon hybrid likewise misses the true chamber boundaries. SciExplorer misses the

\max(x,0)

terms. Allowing separate formulas in different regions can improve predictive accuracy, but none of these searches recovers the complete formula found by our research team.

prompt condition	agent configuration	first page
true hint	Codex 5.5 xhigh	F.1
no hint	Claude Opus 4.8 max	F.2
no hint	Claude Opus 4.8 ultra	F.3
no hint	Codex 5.5 xhigh	F.4
no hint	Fugu ultra	F.5

Appendix

Table 3: Index of the five complete run records included below.

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Machine Collective Intelligence for Explainable Scientific Discovery

Gyoung S. Na, Chanyoung Park

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Deriving governing equations from empirical observations is a longstanding challenge in science. Although artificial intelligence (AI) has demonstrated substantial capabilities in function approximation, the discovery of explainable and extrapolatable equations remains a fundamental limitation of modern AI, posing a central bottleneck for AI-driven scientific discovery. Here, we present machine collective intelligence, a unified paradigm that integrates two fundamental yet distinct traditions in computational intelligence--symbolism and metaheuristics--to enable autonomous and evolutionary discovery of governing equations. It orchestrates multiple reasoning agents to evolve their symbolic hypotheses through coordinated generation, evaluation, critique, and consolidation, enabling scientific discovery beyond single-agent inference. Across scientific systems governed by deterministic, stochastic, or previously uncharacterized dynamics, machine collective intelligence autonomously recovered the underlying governing equations without relying on hand-crafted domain knowledge. Furthermore, the resulting equations reduced extrapolation error by up to six orders of magnitude relative to deep neural networks, while condensing 0.5-1 million model parameters into just 5-40 interpretable parameters. This study marks an important shift in AI toward the autonomous discovery of principled scientific equations.

Abstract

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