A-SR: Self-Evolving Agentic LLMs for Symbolic Regression via Hierarchical Coordination

Imagine you are a detective trying to solve a mystery, but instead of fingerprints or alibis, your clues are numbers on a spreadsheet. Your goal is to find the hidden "recipe"—a neat, written-down math equation—that explains how those numbers behave. This is the world of Symbolic Regression. Unlike standard computer programs that just crunch numbers to give an answer, symbolic regression tries to write the actual law of nature, like $F=ma$ or the formula for gravity, in a way a human can read and understand. It's the difference between a black box that predicts the weather and a meteorologist who can explain why the storm is coming.

For a long time, scientists have used computers to hunt for these recipes, but the process is often clumsy. It's like trying to fix a broken watch by randomly swapping out gears and hoping it starts ticking. If the watch doesn't work, the computer just gets a "bad score" and tries again, often making the same mistakes over and over. Recently, scientists started using Large Language Models (LLMs)—the super-smart AI chatbots you might know—to help write these equations. These AIs are great at guessing what words (or math symbols) should come next, but they can still get stuck, write equations that don't make sense, or get confused when the data gets tricky. The big question is: How do we get these AI detectives to stop guessing randomly and start working like a real team of experts?

Enter A-SR, a new method that treats the search for math formulas not as a solo act, but as a well-orchestrated team effort. The researchers behind this paper, Wenxiao Zhao and their colleagues, realized that the old way of doing things was too rigid. In the past, an AI would just propose an equation, get a score, and try again. If the equation was broken, the AI might try to fix it by adding more numbers, or if it was too simple, it might try to make it more complex, all without really understanding why it failed.

A-SR changes the game by giving the AI a "manager" and a team of specialists. Imagine a newsroom where the Editor-in-Chief (the Coordinator) doesn't just ask one reporter to write a story. Instead, the Editor looks at the day's events and decides which specialist is needed. If the story is full of errors, the Editor calls the Reviewer to check the facts. If the story is boring and needs a new angle, the Generator is called to brainstorm wild ideas. If the story is too long and confusing, the Simplifier is brought in to cut the fluff. And if the story is missing a key piece of the puzzle, the Analyst digs into the archives to find the missing link.

What makes A-SR special is that this team doesn't just follow a fixed schedule. The "Editor" is self-evolving. It watches how the search is going in real-time. If the team is stuck in a loop of making mistakes, the Editor switches the strategy, perhaps telling the team to stop exploring new ideas and focus on fixing the broken ones. It also keeps a "memory" of what worked and what didn't, routing that information to the right specialist. For example, if the team keeps failing because of a specific type of math error, that failure is sent to the Reviewer, who then teaches the team how to avoid it next time.

The paper shows that this approach works incredibly well. When tested on a variety of scientific puzzles—from how chemicals react to how materials stretch under pressure—A-SR found the correct mathematical "recipes" much more often than previous methods. In one set of tests, using a model called Llama3.1-8B, the success rate jumped from about 25.79% to 48.30%. That's nearly doubling the number of problems the AI could solve correctly. Even more impressively, when they took the lessons learned from these team efforts and taught them to a smaller, open-source AI model (Qwen3-4B), that smaller model's success rate also jumped significantly, from 24.58% to 38.29%.

The researchers also tested A-SR on real-world scientific data, like the growth of bacteria and the vibration of mechanical systems. In these tricky, noisy real-life scenarios, A-SR managed to find the best mathematical descriptions in 7 out of 8 cases, beating other top methods. The key takeaway isn't just that the AI got better at math; it's that the AI learned how to think about its mistakes. Instead of blindly trying millions of random combinations, it learned to diagnose its own failures, switch roles when necessary, and use its memory to avoid repeating the same errors. It's a shift from a chaotic solo search to a smart, coordinated team of agents, each doing exactly what they are best at, guided by a manager that knows when to push for new ideas and when to pause and fix what's broken.