Uncovering Physics with AI: The Rise of Equation Discovery

Author: Denis Avetisyan

Symbolic regression is emerging as a powerful tool for automatically identifying underlying mathematical relationships from data, promising a new era of scientific insight.

This review details the application of symbolic regression techniques to automate the discovery of physical laws, build efficient emulators, and advance scientific modeling.

Despite the long-standing goal of automated scientific discovery, extracting interpretable mathematical relationships from data remains a significant challenge. This Special Issue, ‘Introduction to Symbolic Regression in the Physical Sciences’, addresses this by showcasing the emerging field of symbolic regression (SR) – a technique capable of simultaneously identifying both structure and parameters in data-driven models. SR offers a compelling route to uncovering fundamental laws, constructing compact emulators, and deriving effective theories across diverse physical domains. As SR rapidly matures, can we further integrate domain knowledge and theoretical constraints to unlock its full potential for accelerating scientific progress?

The Evolving Landscape of System Modeling

Conventional regression techniques, while widely applied, encounter significant limitations when analyzing datasets exhibiting intricate, non-linear dynamics. These methods typically assume relationships can be adequately represented by straight lines or simple curves, necessitating substantial pre-processing and feature engineering to transform the data into a suitable format. This process demands considerable domain expertise to identify relevant variables and interactions, often requiring researchers to manually construct terms – such as polynomial features or interaction effects – to capture the underlying complexity. Consequently, the success of traditional regression heavily relies on the user’s ability to accurately represent the system’s behavior through these crafted features, and can fail to capture hidden relationships if these features are poorly chosen or incomplete. The reliance on human intuition and pre-defined structures limits the ability of these methods to automatically discover and model complex phenomena directly from raw data, hindering the exploration of potentially novel or unexpected relationships within the dataset.

Conventional regression techniques, while adept at identifying correlations, often fall short when tasked with revealing the fundamental principles governing a system. The resulting models, frequently expressed as complex, opaque equations, provide limited insight into why a phenomenon occurs, focusing instead on merely what happens. This lack of interpretability poses a significant challenge to scientific advancement, as researchers are left with predictive tools devoid of mechanistic understanding. For example, a regression model might accurately forecast the trajectory of a projectile, but fail to explicitly represent the influence of gravity or air resistance – crucial elements for building a comprehensive physical model. Consequently, these “black box” approaches impede the ability to generalize findings, make informed predictions under novel conditions, or ultimately, deepen scientific knowledge beyond the specific dataset used for training.

The limitations of conventional regression techniques in capturing intricate relationships within data have motivated the emergence of Symbolic Regression (SR) as a powerful alternative. Unlike traditional methods that seek to minimize error based on predefined model structures, SR automatically searches for mathematical expressions that best describe the underlying data, effectively discovering governing equations without prior assumptions. This data-driven approach circumvents the need for extensive feature engineering and domain expertise, allowing researchers to uncover potentially unknown physical laws directly from observations. By representing models as symbolic expressions – such as $y = ax^2 + bx + c$ – SR provides interpretable and transparent results, fostering a deeper scientific understanding beyond mere predictive accuracy. The technique effectively bridges the gap between black-box modeling and the desire for explicit, generalizable equations that capture the fundamental principles governing a system.

The Genesis of Discovery: Genetic Programming and Its Descendants

Genetic Programming (GP) is a search-based approach to symbolic regression (SR) that iteratively evolves a population of candidate mathematical expressions. These expressions, typically represented as tree structures, are subjected to processes analogous to biological evolution. Selection favors expressions that minimize error when fitted to training data. Crossover combines parts of two parent expressions to create offspring, exploring new combinations of functions and terminals. Mutation introduces random changes to individual expressions, maintaining diversity and potentially discovering novel solutions. Through repeated cycles of these operations, GP aims to discover equations that accurately model the relationships within a given dataset. The performance of GP is directly related to the choice of terminal set (constants and variables), function set (mathematical operators), and the fitness function used to evaluate candidate solutions.

Modern symbolic regression implementations such as PySR, PyOperon, and AI Feynman build upon the foundations of Genetic Programming (GP) by incorporating several performance-enhancing strategies. These include the use of modern optimization algorithms – often gradient-based methods combined with GP – to accelerate the search for optimal equations. Furthermore, these implementations extensively utilize parallel processing, distributing the computational workload across multiple CPU cores or GPUs to significantly reduce execution time. PySR, for example, incorporates techniques like automatic differentiation and just-in-time compilation to improve efficiency, while AI Feynman leverages a multi-stage evolutionary process and equation simplification routines. These advancements address the computational limitations of naive GP, enabling the discovery of complex relationships from data in a more practical timeframe.

Standard Genetic Programming (GP) implementations can exhibit significant computational expense due to the combinatorial nature of the search space. This is further exacerbated by the phenomenon of “bloat,” where evolved equations grow unnecessarily complex without a corresponding improvement in accuracy. Such bloat increases computational time, often by a factor of 10 or greater, as more complex terms require more calculations during evaluation and fitness assessment. Strategies to mitigate bloat and improve search efficiency include parsimony pressure – penalizing equation complexity – and the use of dimensionality reduction techniques to simplify the search space. Furthermore, employing efficient code generation and parallel processing can reduce the overall time required for GP execution.

Bridging the Gap: Deep Learning and Intelligent Equation Discovery

Equation discovery systems like EQL and uDSR leverage a combined deep learning and reinforcement learning (RL) approach to optimize the symbolic regression (SR) process. Deep learning models are utilized to predict promising equation structures or components, effectively reducing the search space. These predictions then inform an RL agent, which acts as a policy to select which equations to evaluate next. The agent receives rewards based on the accuracy of the generated equations, iteratively refining its search strategy. This integration accelerates convergence – the speed at which a satisfactory equation is found – and improves solution quality, measured by metrics such as $R^2$ and root mean squared error (RMSE), compared to traditional SR methods that rely on random or genetic algorithms.

AI Descartes distinguishes itself in symbolic regression by incorporating axiomatic and formal proof methods into the equation discovery process. This approach moves beyond purely data-driven searches by establishing a framework of logical rules and constraints, allowing the system to not only identify equations that fit the observed data but also to verify their mathematical validity. Specifically, Descartes utilizes formal proof techniques to rigorously demonstrate that derived equations are consistent with established mathematical principles and axioms. This results in solutions that are demonstrably correct, rather than merely approximations, and offers improved interpretability by providing a traceable derivation path for each discovered equation, unlike black-box deep learning approaches. The system’s output therefore prioritizes mathematical certainty and transparency in addition to predictive accuracy.

SHRED and BRUSH represent advancements in symbolic regression optimization. SHRED employs sparse regression techniques, prioritizing simpler models by penalizing coefficient magnitude during the model fitting process. This approach reduces overfitting and improves generalization performance. BRUSH, conversely, utilizes multi-objective optimization, simultaneously optimizing for multiple criteria, such as minimizing error and model complexity, typically measured by the number of operations or terms. This results in a Pareto front of solutions, allowing users to select the optimal trade-off between accuracy and interpretability. Both methods refine the SR process by focusing the search on more promising areas of the solution space and generating more efficient and understandable models.

Amplifying Efficiency and Robustness: Techniques for Complex Systems

Symbolic Regression (SR) often involves evaluating a vast number of mathematical expressions, making computational efficiency paramount. Techniques like Zobrist Hashing and Equality Graphs address this challenge by cleverly identifying and discarding redundant calculations. Zobrist Hashing assigns a unique identifier to each expression, allowing the system to quickly determine if an equivalent expression has already been assessed – avoiding repetitive computations. Equality Graphs take a more structural approach, representing expressions as nodes in a graph, where equivalent expressions are linked, effectively consolidating the search space. These methods drastically reduce the overall computational burden, particularly when dealing with complex datasets or lengthy equations, enabling SR to tackle problems previously considered intractable and accelerating the discovery of underlying relationships within the data.

Symbolic Regression (SR) is increasingly leveraging the power of Foundational Models and Large Language Models (LLMs) to overcome traditional limitations in both hypothesis generation and result interpretation. These models, pre-trained on vast datasets of scientific literature and code, can propose plausible mathematical expressions as initial SR starting points, effectively reducing the search space and accelerating the discovery process. Furthermore, LLMs are capable of analyzing the resulting equations, providing human-readable explanations of their structure and identifying potential relationships between variables – a crucial step for validating findings and ensuring scientific rigor. This synergy between machine learning and SR not only enhances the speed of equation discovery but also improves the accuracy and interpretability of the models generated, facilitating a deeper understanding of the underlying phenomena being studied and enabling more effective applications of the discovered relationships, such as in $y = ax^2 + bx + c$.

Symbolic Regression (SR) doesn’t simply seek equations that fit data; it actively favors simplicity, guided by the principle of Parsimony and formalized through techniques like Minimum Description Length (MDL). MDL posits that the best model isn’t necessarily the most complex one achieving a given level of accuracy, but rather the most concise representation – the equation requiring the least information to describe both its structure and its predictive power. This prioritization is crucial because overly complex equations, while potentially achieving marginally better fits to the training data, often suffer from overfitting and generalize poorly to new observations. Moreover, a simpler equation – one with fewer terms and lower polynomial degree – is inherently more interpretable, offering valuable insight into the underlying relationships driving the observed phenomenon. The result is not just a predictive tool, but a pathway to deeper scientific understanding, allowing researchers to extract meaningful, actionable knowledge from complex datasets and express it in a readily understandable $y = f(x)$ form.

Forecasting the Future: Symbolic Regression and the Advancement of Science

Symbolic Regression (SR) is rapidly becoming a powerful tool in scientific discovery, offering a unique approach to modeling complex phenomena. Unlike traditional methods that require researchers to predefine the form of an equation, SR algorithms autonomously search for mathematical expressions that best fit observed data. This data-driven approach allows for the identification of underlying physical laws without a priori assumptions about their structure. By directly analyzing experimental results, SR can uncover relationships and equations – such as $E=mc^2$ – that might otherwise remain hidden, accelerating breakthroughs in fields ranging from physics and chemistry to biology and engineering. The ability to automatically derive interpretable equations from data represents a significant shift, empowering scientists to explore complex systems and generate novel hypotheses with unprecedented efficiency.

The creation of Emulators represents a significant advancement facilitated by Symbolic Regression. These Emulators are essentially compact mathematical expressions, derived from complex data through SR, that faithfully mimic the behavior of computationally expensive simulations. Rather than running lengthy simulations for each new set of input parameters, researchers can rapidly evaluate the Emulator – often a simple equation like $y = ax^2 + bx + c$ – to obtain accurate predictions. This speed advantage unlocks the potential for extensive parameter space exploration, allowing scientists to efficiently identify critical variables, optimize designs, and ultimately gain deeper insights into the underlying phenomena being modeled. The ability to swiftly assess numerous scenarios, previously intractable due to computational limitations, positions Emulators as a powerful tool for accelerating scientific discovery across diverse fields.

The pursuit of automated scientific discovery is rapidly advancing through innovative symbolic regression techniques, notably Exhaustive Symbolic Regression and the Bayesian Machine Scientist. Exhaustive SR rigorously explores all possible mathematical expressions fitting provided data, guaranteeing the identification of the simplest, most accurate model-a process often exceeding the capabilities of human analysts. Complementing this, the Bayesian Machine Scientist employs Bayesian optimization to intelligently search the space of potential equations, iteratively refining models based on their predictive power and complexity. This approach not only identifies governing equations but also assesses the uncertainty associated with them, providing a more robust understanding of the underlying phenomena. These cutting-edge methods promise to accelerate the pace of scientific progress by automating hypothesis generation and validation, ultimately enabling breakthroughs in fields ranging from materials science to drug discovery, and potentially revealing previously unknown physical laws described by compact mathematical forms like $E=mc^2$.

The pursuit of empirical models, as detailed in the exploration of symbolic regression, echoes a fundamental truth about systems: they learn to age gracefully. G.H. Hardy observed, “A mathematician, like a painter or a poet, is a maker of patterns.” This creation of patterns, whether through mathematical proof or machine learning algorithms, isn’t about achieving a perfect, unchanging state, but rather about revealing the underlying structure within a dynamic system. Symbolic regression, in its quest to rediscover physical laws, doesn’t aim to stop the inherent decay of models, but to understand and emulate the patterns that emerge over time. Sometimes, observing the process of discovery is more valuable than simply seeking a final, definitive answer.

What Lies Ahead?

The pursuit of automated equation discovery, as exemplified by symbolic regression, is not a search for immutable truth, but a mapping of transient system behavior. Each identified relationship, each discovered ‘law’, is merely a locally accurate description, subject to eventual decay as conditions shift. The field’s current emphasis on model accuracy-reducing error metrics-risks obscuring the more fundamental challenge: understanding the limits of those models and anticipating their failure modes. Time is not a metric for improvement, but the medium within which all approximations erode.

Future work must address the inherent brittleness of these discovered relationships. Robustness to noise and extrapolation beyond the training domain remain significant hurdles. A fruitful direction lies in explicitly incorporating uncertainty quantification into the symbolic regression process, acknowledging that every equation carries an implicit expiration date. The creation of ‘self-aware’ models-those capable of signaling their own diminishing validity-would represent a genuine advance.

Ultimately, the value of symbolic regression isn’t simply in generating equations, but in accelerating the cycle of system understanding, failure, and refinement. Incidents are not deviations from a desired state, but essential steps toward maturity. The field’s long-term trajectory hinges on embracing this perspective – viewing discovered models not as final products, but as temporary scaffolding in an ongoing process of adaptation.

Original article: https://arxiv.org/pdf/2512.15920.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/