⚛️ high-energy theory

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

This paper demonstrates that a multi-agent AI workflow, where a lead principal investigator coordinates complementary analytic and numerical tasks for research agents, successfully rediscovered a complex geometric formula for nonlinear surface-wave scattering and discovered a new verified analytic expression for a six-point amplitude, outperforming both single-prompt LLMs and conventional machine learning methods.

Original authors: Zihan Zhou, Digvijay Wadekar, Matias Zaldarriaga

Published 2026-10-02

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Zihan Zhou, Digvijay Wadekar, Matias Zaldarriaga

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a vast, restless ocean where waves do not simply pass through one another but collide, merge, and exchange energy in complex patterns. When these waves are small, their behavior is predictable and simple, like adding numbers together. But when they grow larger, they interact in a nonlinear way, creating a chaotic dance of energy that defies simple addition. Physicists have long sought a single, elegant mathematical rule that could describe the strength of these interactions for any number of waves, no matter how they are arranged. The challenge is that the rules change depending on the specific frequencies of the waves involved. In one set of conditions, a simple formula works perfectly; in a neighboring set of conditions, a completely different formula takes over. The true goal is to find one master expression that can seamlessly switch between these different rules, covering every possible scenario without ever breaking down.

For decades, this problem remained a puzzle, requiring researchers to manually piece together different formulas for different regions of the wave spectrum. Recently, a breakthrough occurred when a human researcher and an artificial intelligence agent worked together to discover a unified expression, which they named the "hydrotope." This new formula acts like a geometric map, stitching together all the separate rules into a single, global description of how water waves scatter. However, the story of how they found it remained a black box. Did the AI truly understand the physics, or did it just get lucky? To answer this, a team of researchers at Princeton University, the University of Texas at Austin, and the Institute for Advanced Study set out to reverse-engineer the discovery. They wanted to see if an AI could find this complex formula on its own, or if it needed a human hand to guide it through the maze of changing rules.

The researchers began by testing the AI in a controlled environment. They gave the computer access to a powerful tool that could calculate the exact outcome of wave collisions for any specific set of frequencies, but they did not give it the answer. They asked the AI to look at the data and figure out the underlying formula. They ran this experiment eighteen times, using different types of AI models and giving them different levels of help. In some runs, the AI was given a false hint, told to look for a simple ratio of polynomials that would work everywhere. In others, it was given a true hint, told that different rules apply in different regions. In the final set of runs, the AI was given no hints at all, left to discover the pattern from scratch.

The results were revealing. Most of the AI agents, working alone, hit a wall. They were quite good at finding the correct formula for a single, specific region of wave frequencies. They could look at the data, see the pattern, and write down a rule that worked perfectly for that one case. However, they consistently failed to realize that the rule needed to change when the conditions shifted. They would find the right answer for one part of the puzzle but stop there, unable to combine the different pieces into the single, global formula that covers all possibilities. Even when given a false hint that led them down the wrong path, many agents stubbornly tried to force the data to fit that incorrect idea, rather than abandoning it when the numbers didn't add up. Only a handful of the single-agent runs managed to piece together the complete picture, and even then, they often missed the final steps needed to verify that their formula worked in every possible scenario.

The researchers realized that the problem wasn't that the AI couldn't do the math; it was that the AI was trying to do too much at once. It was acting as both the architect, designing the formula, and the inspector, checking if it was correct. When the inspector was the same mind as the architect, it tended to look only at the evidence that supported the design and missed the flaws. To fix this, the team designed a new workflow inspired by how human research teams operate. They created a system with a "principal investigator" and two "students." The two students worked in parallel, each taking a different approach to the problem. One focused on deriving the formula, while the other focused on testing it in new, difficult situations. The principal investigator acted as a coordinator, reading their work, spotting the gaps, and assigning new tasks to fill those gaps. Crucially, the principal investigator ran independent checks using code written from scratch, ensuring that the final answer was not just a lucky guess but a verified truth.

This team approach changed everything. When the researchers ran the same difficult problem with this coordinated group, the result was immediate and complete. The team successfully rediscovered the full hydrotope formula without any human hints, finding the correct rule for every single region and verifying it across hundreds of test cases. The system didn't just find a local pattern; it understood the global structure. It learned to look for the boundaries where the rules changed and to stitch the different pieces together into a seamless whole. The researchers then pushed the system further, asking it to solve an even harder problem: finding the formula for a more complex type of wave interaction involving three waves moving in one direction and three in another. This was a problem for which no known formula existed. The team succeeded again, producing a new, verified mathematical expression for this six-wave interaction that had never been seen before.

The study highlights a fundamental shift in how artificial intelligence can contribute to science. It shows that while AI models are powerful tools for calculation and pattern recognition, they often struggle with the higher-level task of synthesizing a complete theory from partial truths. They can find the right answer for a specific case but fail to see the bigger picture. By separating the roles of discovery and verification, and by having different agents challenge each other's work, the researchers created a system that could overcome these limitations. The AI didn't just mimic human discovery; it replicated the collaborative process that makes human science work. The findings suggest that the future of scientific discovery with AI may not lie in building a single, super-intelligent agent that does everything, but in building teams of specialized agents that work together, check each other's work, and push past the limits of what any one of them could achieve alone. The hydrotope was found, and a new formula for wave interactions was born, not by a lone genius, but by a coordinated effort that turned a collection of partial answers into a complete truth.

Technical Summary: How AI Agents Discover Scientific Equations

Problem Statement

The paper investigates the capabilities and limitations of AI agents in discovering complex scientific formulas, specifically focusing on the hydrotope, a geometric formula describing nonlinear surface-wave scattering in deep water. The core challenge lies in the chamber-dependent nature of the solution.

In one-dimensional deep-water scattering, the $n$ -point amplitude ( $A_n$ ) is not described by a single polynomial across the entire frequency space. Instead, the frequency space is divided into "chambers" (regions defined by specific inequalities of input frequencies). Within each chamber, the amplitude follows a specific polynomial. Crossing a boundary (where a sum of squared frequencies equals a minimum squared frequency) switches the active terms on or off. The global solution requires combining exponentially many potential polynomial pieces into a single analytic expression using the max(x, 0) (or $(x)_+$ ) function to handle these switches.

The specific task is to derive the all- $n$ formula for the "two-minus" sector (where two wavenumbers are negative) and, subsequently, a new formula for the "three-minus" sector at six points ( $A_6$ ). The agents have access to exact numerical values generated via Berends–Giele (BG) recursion but must infer the analytic structure from these discrete points.

Methodology

The study employs a controlled experimental design comparing three distinct approaches:

1. Single-Agent Rediscovery (18 Runs)

The authors reconstructed the original human-agent discovery process and repeated it under three prompt conditions using six different LLM configurations (e.g., Claude Opus, Codex, DeepSeek):

No Hint: The agent receives only the physical setup and BG code.
False Hint: The agent is told the answer is a single rational function (ratio of polynomials) valid everywhere, with advice to avoid boundary-crossing frequencies.
True Hint: The agent is told the answer is a piecewise homogeneous polynomial where different polynomials apply in different chambers, without revealing the boundaries or the formula.

2. Baseline Comparisons

The agent-discovered formulas are compared against:

Symbolic Regression (SR): Conventional methods (PySR, gplearn, Operon) and LLM-assisted SR (LaSR, LLM-SR).
Machine Learning (ML): Standard regressors (Linear, Random Forest, MLP) and Kolmogorov–Arnold Networks (KAN).
Metric: Performance is measured by Held-out Mean Absolute Error (MAE) on points not used for training, specifically testing whether methods can recover the exact analytic form or merely approximate the data.

3. Coordinated Multi-Agent System (PI + Two Students)

To address the failure modes of single agents, the authors designed a workflow mimicking a research team:

Roles: A Principal Investigator (PI) coordinates the effort, while two Student Agents pursue complementary tasks (e.g., one focuses on deriving the formula, the other on numerical testing).
Process:
1. Students propose formulas and run tests, recording results in a shared document.
2. The PI independently implements the proposed formula in separate code and tests it against exact BG calculations in new, unsampled chambers.
3. Failures in the PI's independent tests become specific tasks for the next round.
Application: This workflow was applied to rediscover the hydrotope (two-minus) and to discover a new formula for the six-point three-minus amplitude ( $A_6$ ).

Key Results

Single-Agent Performance

Success Rate: Only 4 out of 18 single-agent runs successfully recovered the complete hydrotope formula. All four successful runs occurred under the True Hint condition.
Failure Modes:
- Local vs. Global: Most agents (15/18) correctly identified the formula within a single "principal" chamber but failed to combine these into a global expression.
- Boundary Detection: Agents struggled to identify all chamber boundaries (sums of frequencies) and the alternating subset sum structure required to combine them.
- Hint Sensitivity: Under the False Hint, agents either stuck to the incorrect rational function form or rejected it but failed to find the correct piecewise alternative.
- Testing Bias: Agents often tested formulas only on configurations similar to those used for derivation, missing edge cases where terms should switch on/off.

Baseline Comparisons

Approximation vs. Discovery: While some SR and ML methods achieved low average error (e.g., Operon, Random Forest), none recovered the exact analytic formula. They failed to capture the max(x, 0) switching behavior and the specific chamber boundaries, resulting in non-zero error on held-out data.
Limitation: Standard SR minimizes average error on a fixed dataset, which allows it to miss rare chamber transitions that occur in sparsely sampled regions.

Multi-Agent System Performance

Hydrotope Rediscovery: Both PI + Two Students teams (using Claude Opus and Codex 5.5) successfully rediscovered the complete hydrotope formula without hints, achieving zero error on extensive tests.
New Discovery ( $A_6$ ): Applied to the harder three-minus sector, the team discovered a new, verified analytic expression for the six-point amplitude $A_6$ $A_{6}$ .
- The formula passed 140/140 exact tests across 58 chambers.
- The workflow identified that corrections involving pairs of chamber boundaries were necessary, a detail missed by initial attempts.
Comparison of Architectures:
- Single Agent: Failed to produce a global formula.
- PI + 1 Student: Found formulas for two opposite chambers but failed to combine them into a single expression valid for a third chamber.
- PI + 2 Students: Successfully separated formula construction from cross-chamber testing, allowing parallel progress and independent verification.

Significance and Claims

The paper claims that the primary bottleneck in AI-driven scientific discovery is not finding local patterns but synthesizing global structures and rigorously testing edge cases.

The Necessity of Coordination: The study demonstrates that separating the roles of "hypothesis generation" and "independent verification" is crucial. A single agent tends to conflate the data used to build a model with the data used to test it, leading to overfitting on specific chambers. A coordinated team with a dedicated verifier (PI) can identify missing cases and boundary conditions that a single agent misses.
Beyond Fitting: The results highlight a distinction between predictive accuracy and scientific discovery. Standard ML and SR methods can approximate the data but fail to recover the underlying analytic structure (the "chamber walls") because they optimize for average error rather than physical consistency across boundaries.
The Role of Counterexamples: The success of the multi-agent system relies on using exact counterexamples (failures in new chambers) to iteratively revise the formula. This "propose–test–revise" cycle, driven by a dedicated verifier, allows the system to navigate the exponential complexity of the solution space.
Scientific Contribution: The paper provides a new, independently verified analytic formula for the six-point three-minus scattering amplitude ( $A_6$ ), which was previously unknown. It also establishes a benchmark for evaluating AI agents on tasks requiring the discovery of piecewise analytic functions.

The authors conclude that while general-purpose AI models are powerful, their effectiveness in scientific discovery is significantly enhanced by structured workflows that enforce independent verification and parallel exploration of different aspects of the problem.