Uncovering Nature’s Code: A Stepwise Approach to Physical Law Discovery

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Author: Denis Avetisyan

Researchers have developed a new method that mimics how scientists build understanding, progressively revealing underlying physical laws from data.

Through analysis of both planetary and binary star systems, a computational framework successfully rediscovered Kepler’s third law - initially expressed with differing formulations dependent on system mass ratios - and ultimately synthesized these findings into a generalized, unified statement of the universal law of gravitation, [latex]F=f(M,m,T,R)[/latex], demonstrating a pathway for discovering fundamental physical laws from observational data.
Through analysis of both planetary and binary star systems, a computational framework successfully rediscovered Kepler’s third law – initially expressed with differing formulations dependent on system mass ratios – and ultimately synthesized these findings into a generalized, unified statement of the universal law of gravitation, [latex]F=f(M,m,T,R)[/latex], demonstrating a pathway for discovering fundamental physical laws from observational data.

This paper introduces Chain of Symbolic Regression (CoSR), a hierarchical modeling framework that improves upon traditional symbolic regression by learning physical laws in a stepwise fashion from simple to complex relationships.

Conventional symbolic regression, while powerful for knowledge discovery, often struggles with complex physical systems, yielding lengthy and physically meaningless expressions. This limitation arises from its failure to mirror the hierarchical, progressive nature of scientific discovery-physical laws evolve from simplicity to complexity. Addressing this, our work, ‘Data-driven Progressive Discovery of Physical Laws’, introduces Chain of Symbolic Regression (CoSR), a novel framework that models the discovery of physical laws as a chain of knowledge, progressively combining meaningful units to reveal underlying principles. Demonstrated across problems ranging from fluid dynamics to laser-metal interaction-and even uncovering new scaling laws in aerodynamics-CoSR offers a pathway to more interpretable and accurate data-driven scientific models; but can this approach unlock entirely new physical insights beyond refinement of existing theories?

The Limits of Linear Thinking

Many challenges in science and engineering stem from systems where effects aren’t proportional to their causes – a phenomenon known as non-linearity. Unlike simple, linear models where a straight line can predict outcomes, these complex systems exhibit intricate relationships, meaning a small change in one variable can trigger disproportionately large, and often unpredictable, consequences. Consider weather patterns, fluid dynamics, or even the spread of diseases; each involves countless interacting factors where traditional modeling techniques – relying on averaged values and simplified assumptions – often fall short. These methods struggle to capture the emergent behaviors and feedback loops inherent in non-linear systems, leading to inaccuracies and a limited understanding of the underlying processes. Consequently, researchers are increasingly seeking advanced computational approaches capable of navigating these complexities and uncovering the hidden dynamics within seemingly chaotic systems.

Conventional modeling techniques frequently demand researchers articulate precise, pre-defined relationships within a system, necessitating substantial manual feature engineering to translate raw data into usable inputs. This process isn’t merely about data preparation; it inherently biases the analysis toward pre-conceived notions of how the system functions, effectively limiting the potential for genuinely novel discoveries. The requirement for strong prior assumptions-essentially, telling the model what to look for-can inadvertently obscure the subtle, unexpected patterns that might reveal deeper underlying principles. Consequently, while these traditional methods can yield results, they often function as confirmation tools rather than exploratory ones, hindering a full understanding of complex phenomena and restricting the generalizability of any findings to scenarios closely mirroring the initial assumptions.

The pervasive need for prior assumptions in many modeling approaches introduces a critical limitation on scientific understanding. When researchers impose constraints based on existing beliefs, the resulting models may inadvertently mask the true, underlying principles governing a system. This isn’t simply a matter of inaccuracy; it actively hinders discovery by steering investigations towards pre-conceived notions rather than allowing emergent patterns to reveal themselves. Consequently, the generalizability of findings suffers, as models built on specific assumptions may fail when applied to slightly different scenarios or datasets. A model tightly coupled to initial beliefs, however accurate within those constraints, offers a limited, and potentially misleading, view of the broader reality, hindering the development of truly universal and predictive frameworks.

Symbolic regression enables knowledge discovery through three strategies: hierarchical modeling of composite functions, identification of implicit relationships, and simplification of expressions via mathematical transformations.
Symbolic regression enables knowledge discovery through three strategies: hierarchical modeling of composite functions, identification of implicit relationships, and simplification of expressions via mathematical transformations.

Automating Equation Discovery

Symbolic regression distinguishes itself from traditional regression methods by not requiring a pre-defined model structure. Instead of fitting data to a specified equation – such as a linear or polynomial function – symbolic regression utilizes algorithms to directly search the space of possible mathematical expressions. This search is typically conducted using genetic programming or similar evolutionary techniques, evaluating the fitness of each candidate equation based on its ability to accurately predict the observed data. The process yields an equation, expressed in mathematical terms with variables and operators [latex] f(x) = ax^2 + bx + c [/latex], that best describes the relationship within the dataset without prior assumptions about its functional form. This approach is particularly useful when the underlying relationship between variables is unknown or complex, and traditional modeling techniques may fail to capture the true patterns.

Symbolic regression distinguishes itself from traditional regression methods by producing explicit mathematical equations, not merely predictive models. While standard regression aims to minimize error between predicted and actual values, symbolic regression simultaneously searches for a mathematical expression – involving variables, constants, and mathematical operators – that best fits the data. This resulting equation, such as [latex]y = ax^2 + bx + c[/latex], provides a directly interpretable relationship between input and output variables, enabling users to understand how the prediction is made and gain insights into the underlying data generating process. The interpretability facilitates validation, allows for extrapolation beyond the training data, and supports the discovery of previously unknown relationships.

Symbolic regression differentiates from conventional modeling techniques by not requiring a pre-defined model structure. Traditional methods, such as linear or polynomial regression, necessitate the user to specify the functional form of the relationship being modeled. Symbolic regression, however, explores a vast search space of potential equations – including various mathematical operations and functions – using algorithms like genetic programming. This allows it to discover non-obvious relationships and functional forms within the data that might be overlooked when constrained by a pre-defined model, particularly in cases involving complex, non-linear interactions or when the underlying equation is unknown. The technique effectively performs model discovery rather than model fitting, potentially revealing previously unknown or unanticipated mathematical relationships represented by equations such as [latex]y = ax^2 + bsin(x)[/latex].

The Chain of Symbolic Regression (CoSR) framework systematically discovers physical knowledge by progressively reducing dimensionality, compressing hierarchical relationships, and refining expressions-guided by principles of parsimony, interpretability, and predictive accuracy-to create a balanced knowledge chain.
The Chain of Symbolic Regression (CoSR) framework systematically discovers physical knowledge by progressively reducing dimensionality, compressing hierarchical relationships, and refining expressions-guided by principles of parsimony, interpretability, and predictive accuracy-to create a balanced knowledge chain.

Refining the Search for Underlying Principles

Hierarchical discovery in symbolic regression operates by decomposing a target relationship into a hierarchy of sub-expressions and intermediate variables. This process reduces the complexity of the search space by allowing the algorithm to first identify and optimize these intermediate components before combining them to model the final relationship. Instead of directly searching for a single, complex equation, the method sequentially builds up expressions, effectively partitioning the problem into smaller, more manageable steps. This approach significantly improves computational efficiency, particularly when dealing with high-dimensional data or intricate relationships, as it reduces the number of possible combinations the algorithm must evaluate. The resulting model, while potentially longer in terms of the number of operations, often exhibits improved generalization performance due to the simplified search and reduced risk of overfitting.

Transformation Discovery operates by applying algebraic manipulations and trigonometric identities to symbolic expressions generated during the regression process. This process aims to represent the same mathematical relationship using a functionally equivalent, but structurally simpler, form. For example, a complex polynomial might be rewritten using fewer terms, or a trigonometric expression could be consolidated using identities like [latex]sin^2(x) + cos^2(x) = 1[/latex]. This simplification directly reduces the computational cost of evaluating the expression, particularly during model validation and prediction. Furthermore, simpler expressions are easier for human analysts to interpret, improving the overall understandability of the derived model and facilitating knowledge discovery.

Implicit Discovery, within the context of symbolic regression, operates by identifying constraints and relationships not explicitly defined in the initial problem formulation or readily apparent in the raw data. This is achieved through algorithms that explore the solution space for functional forms that satisfy these hidden dependencies, effectively reverse-engineering the underlying governing principles. Unlike traditional methods that rely on predefined operators and functions, Implicit Discovery can uncover relationships expressed through complex interactions or non-obvious mathematical constructs. This often involves identifying variables that, while not directly present in the target equation, significantly influence the model’s predictive power when included as intermediate terms or constraints, thereby revealing previously unknown data characteristics and improving model accuracy and generalization.

A hierarchical symbolic regression pathway progressively discovers key dimensionless numbers-Prandtl, Grashof, and Rayleigh-from raw parameters in turbulent Rayleigh-Bénard convection, ultimately refining the conventional Nusselt-Rayleigh relationship into a simplified linear scaling law and revealing underlying physical insights.
A hierarchical symbolic regression pathway progressively discovers key dimensionless numbers-Prandtl, Grashof, and Rayleigh-from raw parameters in turbulent Rayleigh-Bénard convection, ultimately refining the conventional Nusselt-Rayleigh relationship into a simplified linear scaling law and revealing underlying physical insights.

From Data to Insight: Practical Implementation and Dimensional Understanding

PySR represents a significant advancement in the field of symbolic regression, offering a practical and highly effective framework for uncovering underlying mathematical relationships from data. This library distinguishes itself through its robust implementation of various CoSR (constrained symbolic regression) strategies, enabling researchers to incorporate prior knowledge – such as known physical laws or dimensional constraints – directly into the equation discovery process. Unlike traditional symbolic regression methods which often yield complex and physically unrealistic results, PySR streamlines the search for governing equations, focusing computational resources on plausible solutions. The library’s efficiency is further enhanced by its ability to handle large datasets and complex functional forms, making it a valuable tool for scientists and engineers across diverse disciplines seeking to model and understand complex phenomena.

Dimensional analysis, rooted in the Buckingham Pi Theorem, serves as a powerful constraint during symbolic regression, fundamentally improving the reliability and interpretability of discovered equations. This theorem identifies dimensionless groups – combinations of physical variables that remain constant across different units – thereby drastically reducing the complexity of the search space. Instead of exploring all possible mathematical relationships between variables, the process focuses on these dimensionless Pi groups, ensuring that any resulting equation is physically realistic and independent of the chosen unit system. By enforcing dimensional consistency, the technique effectively filters out spurious correlations and prioritizes relationships grounded in fundamental physical principles, leading to more robust and generalizable models. This approach is particularly valuable when dealing with complex systems where the underlying physics are not fully understood, as it provides a framework for extracting meaningful relationships from observational data, even with limited prior knowledge.

The integration of symbolic regression with dimensional analysis demonstrably streamlines the process of uncovering fundamental relationships within complex datasets. By leveraging principles like the Buckingham Pi Theorem, the search space for potential equations is significantly reduced, focusing computational effort on physically plausible solutions. This approach proved particularly effective in studies of laser-metal interaction, where the combined methodology achieved remarkably high accuracy – evidenced by [latex]R^2[/latex] values reaching 0.982. Such results indicate a capacity to not only identify scaling laws but to do so with a level of precision that surpasses traditional methods, suggesting a powerful tool for advancing scientific discovery across various disciplines.

A progressive discovery pathway, combining dimensional analysis and scaling transformations, reveals a parsimonious and unified scaling form for rough-wall pipe flow-reproducing the classic Goldenfeld’s law while achieving superior data collapse and improved prediction accuracy, especially in the transitional turbulent regime, by reconstructing the friction factor [latex]C_f[/latex] using polynomials.
A progressive discovery pathway, combining dimensional analysis and scaling transformations, reveals a parsimonious and unified scaling form for rough-wall pipe flow-reproducing the classic Goldenfeld’s law while achieving superior data collapse and improved prediction accuracy, especially in the transitional turbulent regime, by reconstructing the friction factor [latex]C_f[/latex] using polynomials.

The pursuit of knowledge, as demonstrated by the Chain of Symbolic Regression (CoSR) framework, necessitates a methodical dismantling of complexity. This approach mirrors a fundamental tenet of efficient communication: reduction to essential components. Vinton Cerf observed, “The Internet treats everyone the same.” This equality of treatment, applied to data within CoSR, allows for a progressive build-up of understanding, starting with basic relationships and incrementally layering complexity. The framework’s hierarchical modeling directly addresses the limitations of traditional, one-step symbolic regression by prioritizing clarity through staged discovery. Unnecessary complexity is, indeed, violence against attention; CoSR embodies this principle by focusing on iterative refinement rather than attempting exhaustive, immediate solutions.

Further Refinements

The presented framework, Chain of Symbolic Regression, addresses a persistent challenge: the imposition of simplicity upon complexity. It is not a solution, but a re-framing. The progression from basic to elaborate relationships, while conceptually sound, remains tethered to the initial symbolic space. Future iterations must consider methods for dynamic expansion of the symbolic basis itself – allowing the very language of physical law to evolve alongside the discovered relationships. This necessitates exploration beyond purely numerical data, incorporating qualitative information and existing theoretical constraints to guide the search.

A critical limitation resides in validation. Current metrics assess only predictive accuracy. True discovery requires demonstrably novel predictions, verified through independent experimentation. The field needs robust protocols for distinguishing genuine insight from sophisticated curve-fitting. Clarity is the minimum viable kindness; a predictive model, however elegant, offers little solace without explanatory power.

Ultimately, this work gestures toward a broader question: can automated systems genuinely discover, or merely reproduce? The answer, predictably, is not within the algorithms themselves, but in the careful articulation of what constitutes “discovery”. The pursuit of such a definition, perhaps, is the most valuable outcome.

Original article: https://arxiv.org/pdf/2603.13727.pdf

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

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2026-03-18 06:15