How reinforcement learning and co-design are transforming symbolic regression
Introduction
Symbolic regression has long captivated both researchers and industry practitioners because it attempts to discover the functional form of a relationship directly from data rather than forcing it to fit a preselected equation. In a recent paper, Yuan Tian and colleagues introduces a method called Sym-Q, which weaves together offline reinforcement learning with an interactive “co-design” capability. This co-design approach allows domain experts—physicists, engineers, data scientists, and so on—to guide the symbolic regression process in real time, thus merging human intuition with algorithmic power in a way that can accelerate model discovery.
Their results, published in Nature Communications, show that Sym-Q competes strongly against leading transformer-based approaches and traditional genetic-programming-style methods when tested on standard symbolic regression benchmarks. Beyond simply matching or exceeding previous accuracy levels, Sym-Q reveals its greatest strengths when partial prior knowledge is present: it can “lock in” those expert insights while still exploring uncharted regions of the expression space.
Symbolic regression and its challenges
The fundamental difficulty with symbolic regression is the size and complexity of the equation space. Any given problem might involve standard operations like addition, multiplication, exponentiation, or more specialized transformations such as logarithms and trigonometric functions. Each new operator expands the set of possible expressions, and when more variables and constants are introduced, the space becomes huge. Researchers have pursued several strategies to handle this combinatorial explosion, from genetic programming (iterative evolution of candidate equations) to more recent large-scale transformer models that cast symbolic regression as a language-translation task.
Both approaches have drawbacks. Genetic algorithms may require substantial computational time to evolve solutions, especially if the ultimate equations are complex. Transformer-based architectures rely on teacher-forcing at training time, which can lead to discrepancies when the model is deployed. Error correction at inference becomes difficult because the model is used to “seeing” perfect ground truth tokens. In real environments, there are no ground truth tokens to guide every step, so small missteps can accumulate and derail the final expression.
Co-design and offline RL: What’s new
Sym-Q employs offline reinforcement learning rather than the more common online variant. Instead of having to explore the search space in real time and continually query an environment to receive rewards, the model trains on a large static dataset of “trajectories.” Each trajectory shows how an expression was built step by step, along with a measure of how accurate that expression ultimately turned out to be. By training in this offline manner, Sym-Q compresses lessons learned from thousands (or even millions) of symbolic-regression attempts into a single reusable policy.
The co-design mechanism is perhaps the most intriguing aspect. If a user has partial domain knowledge—for example, they already suspect that the relationship between two variables is additive, or that a friction-like term must appear in a physics equation—Sym-Q can take this partial equation tree as a starting point, then fill in the gaps. This stands in contrast to traditional “one-shot” algorithms that either begin from scratch or attempt to guess the entire form of the equation in a single go. Interactive input can drastically shorten the search, reduce irrelevant exploration, and align the final equations more tightly with known physics or other ground truths.
Numerical experiments and key insights
The authors demonstrate Sym-Q’s performance on the SSDNC dataset, where it recovers the correct symbolic skeleton for a significant fraction of the test equations—often outpacing methods such as NeSymReS and T-JSL. The paper highlights how beam search can be layered on top of Sym-Q to further improve these results, at the cost of some additional computational time. It is notable that even with beam search, Sym-Q’s inference times can be faster than certain transformer-based baselines, thanks to the efficiency gained from offline RL.
The ability to incorporate partial domain knowledge shines in specific scenarios. One set of experiments shows how the model can discover unmodeled “drift” terms in physics equations—factors that represent friction, measurement bias, or other real-world phenomena absent from the classic textbook form. Another experiment examines a synthetic transit-spectra dataset in astrophysics, where Sym-Q initially has trouble finding the correct final expression because the variables span a very broad numerical range. When the authors add minimal hints from domain experts—for instance, the knowledge that one variable has a specific kind of logarithmic dependence—the model quickly refines its search and converges on the correct closed-form solution. This adaptability is precisely the sort of advantage that co-design aims to deliver: it harnesses expert suggestions while allowing the RL mechanism to do the heavier computational lifting.
Final thoughts
Sym-Q takes symbolic regression a step beyond routine equation-finding by letting people and algorithms collaborate in a more flexible, interactive way. If you already know part of the structure underlying a problem or suspect certain terms should appear, the algorithm can integrate that knowledge directly into its search. Because it is trained in an offline RL setting, it does not need to run fresh explorations for every new task, which makes it especially useful for large-scale or time-sensitive industrial settings.
Some open questions remain. The paper focuses mostly on equations with up to three variables, so work remains to see how well this approach will scale to higher-dimensional domains. One also wonders how best to curate the static dataset of expression-building trajectories to maximize the learning potential for offline RL. For now, though, Sym-Q’s results are already promising. They show that merging symbolic regression with the power of reinforcement learning can pay off, and they illustrate what becomes possible when domain expertise is woven into the computational loop rather than sidelined.
In an era when interpretability and trust often matter as much as raw predictive ability, Sym-Q’s success story highlights the enduring appeal of symbolic regression. There is something deeply satisfying about algorithms that not only fit data but also hand you a human-readable formula describing how the world works—a formula that you, as a scientist or engineer, can tweak, validate, and comprehend. If the development of Sym-Q is any indication, we may be on the cusp of a new chapter in AI-driven science, one in which human and machine creativity come together to shape the next generation of scientific insights.