Decoding Data with Symbolic Networks

Author: Denis Avetisyan

A new neural network architecture bridges the gap between deep learning and symbolic regression, offering a path towards interpretable and analytically-recoverable models.

This paper introduces Symbolic-KANs, which embed discrete symbolic structure within a neural network to discover compact analytic expressions from data and constraints.

A persistent challenge in scientific machine learning lies in reconciling the interpretability of symbolic methods with the scalability of neural networks. This work introduces Symbolic–KAN: Kolmogorov-Arnold Networks with Discrete Symbolic Structure for Interpretable Learning, a novel neural architecture that directly embeds discrete symbolic structure within a deep network to bridge this gap. By learning multivariate functions as compositions of univariate primitives, Symbolic-KANs yield compact, closed-form expressions directly from data and governing constraints-without requiring post-hoc symbolic fitting. Could this approach unlock a new paradigm for discovering and representing mechanistic insights from complex systems, moving beyond black-box predictions toward truly interpretable and scalable scientific modeling?

The Illusion of Control: Why Equations Remain Elusive

Historically, discerning the underlying equations that govern natural phenomena from observational data has been significantly constrained by the need for researchers to predefine the mathematical form of those equations. This reliance on prior assumptions – specifying, for instance, whether a relationship is linear, polynomial, or exponential – severely limits the discovery process. While convenient for well-understood systems, this approach falters when confronted with complexity, as it may overlook crucial nonlinear terms or entirely different functional relationships. Consequently, researchers often find themselves constrained by their initial guesses, potentially missing fundamental insights into the system’s true behavior and hindering progress in fields where governing equations remain elusive, such as turbulent fluid flows or the dynamics of novel materials. The pre-specification of functional forms essentially acts as a powerful inductive bias, guiding the search but potentially obscuring the actual governing laws.

Conventional techniques for uncovering the equations governing physical systems frequently falter when confronted with the intricacies of high-dimensional data, particularly in fields like fluid dynamics and materials science. These disciplines often involve a multitude of interacting variables, rendering analytical solutions-those derived through traditional mathematical methods-unattainable. The sheer complexity prevents researchers from directly applying established equation-solving techniques, forcing reliance on computationally expensive simulations or simplified models that may sacrifice crucial details. Consequently, progress is hampered by an inability to efficiently and accurately capture the underlying physics from observational data, motivating the development of novel, data-driven approaches capable of navigating these complex landscapes and revealing hidden governing principles.

The pursuit of understanding complex phenomena increasingly demands a departure from traditional equation discovery methods. Reliance on pre-defined functional forms often proves inadequate when analyzing systems exhibiting non-linear behaviors or operating in high-dimensional spaces, areas common in fields like turbulent flow or advanced materials. A data-driven approach, capable of discerning governing equations directly from observed data, offers a powerful alternative. This methodology doesn’t presume a specific equation structure, instead leveraging algorithms to identify underlying relationships and extrapolate predictive models. Consequently, researchers can explore previously inaccessible regimes, potentially revealing novel physical laws and accelerating innovation across diverse scientific disciplines. The ability to move beyond assumptions and embrace the full complexity of observed data represents a critical step toward unlocking the secrets hidden within intricate systems.

Symbolic-KAN: A Patch, Not a Paradigm Shift

Symbolic-KAN employs a novel neural network architecture designed to integrate discrete symbolic representations directly into the trainable parameters of a deep learning model. Unlike traditional deep networks that output opaque, continuous vector spaces, Symbolic-KAN’s structure allows for the explicit representation of symbolic components – such as terms in an equation – within the network’s layers. This is achieved by defining network parameters not as arbitrary weights, but as coefficients associated with predefined symbolic primitives. The result is a model capable of learning functions and relationships while simultaneously exposing the underlying symbolic form, effectively bridging the interpretability gap often found in black-box neural networks and enabling the direct extraction of human-readable equations or rules from the learned model.

Symbolic-KAN utilizes the Kolmogorov-Arnold Representation Theorem to decompose multivariate functions into a sum of univariate functions, effectively reducing the complexity of the model and enhancing its ability to generalize. This parameterization allows any smooth function of n variables to be represented as a superposition of functions of a single variable, [latex]f(x_1, …, x_n) = \sum_{i=1}^{N} g_i(h_i(x_1, …, x_n))[/latex], where [latex]g_i[/latex] and [latex]h_i[/latex] are univariate functions. By expressing complex relationships in this manner, the model inherently promotes sparsity – many of the component functions [latex]g_i[/latex] will have negligible impact – and improves performance on unseen data by focusing on the essential underlying structure of the function being learned.

Symbolic-KAN enables the identification of governing equations by decomposing complex relationships into a sum of simpler, univariate functions applied to individual input variables. This decomposition, based on the Kolmogorov-Arnold Representation Theorem, allows the network to represent a multivariate function [latex]f(x_1, …, x_n)[/latex] as a series of terms, each of which is a function of a single input [latex]x_i[/latex]. These univariate functions serve as interpretable primitives, and the network learns to combine them with learned coefficients to approximate the original function. The resulting equation, expressed as [latex]\sum_{i=1}^n g_i(x_i)[/latex], offers a transparent and interpretable representation of the learned relationship, allowing for direct examination of the influence of each input variable and facilitating the extraction of governing principles from data.

Forcing Interpretability: A Game of Gates and Penalties

Gated training mechanisms are essential for achieving interpretable results in models utilizing primitive components. Initially, the model learns a weighted combination of these primitives, representing a ‘soft’ selection. Gated training introduces a sigmoid gate applied to the output of each primitive; this gate’s output, ranging from 0 to 1, functions as a selection probability. During training, the gate is optimized to produce discrete selections – values approaching 0 or 1 – effectively choosing which primitives contribute to the final equation. This transformation from soft to discrete combinations is critical because it allows for clear identification of the specific primitives driving the model’s behavior, drastically improving interpretability compared to a blended, continuous representation. The resulting equation then reflects a clearly defined set of selected primitives, rather than a complex, opaque combination of all available options.

Entropy regularization is applied during training to promote sparse selection of primitives within the model. This technique adds a penalty to the loss function proportional to the entropy of the primitive selection distribution; mathematically, this is often expressed as minimizing [latex] -\sum_{i} p_i \log(p_i) [/latex], where [latex] p_i [/latex] represents the probability of selecting the i-th primitive. By encouraging the model to assign high probability to a limited number of primitives and near-zero probability to the rest, entropy regularization simplifies the resulting equation and reduces overfitting. This sparsity improves generalization performance on unseen data by focusing the model on the most salient features and reducing reliance on noisy or irrelevant primitives.

NonMaximumSuppression (NMS) is a post-processing technique applied to the model’s primitive selections to enforce diversity and prevent redundancy. Following the initial selection of primitives, NMS identifies and suppresses highly similar selections based on an intersection-over-union (IoU) threshold. Specifically, for each selected primitive, the algorithm calculates the IoU with all other selected primitives; if the IoU exceeds the defined threshold, the primitive with the lower confidence score is discarded. This process ensures that the final set of selected primitives represents distinct features or components, thereby preventing the model from converging on trivial or repetitive solutions and promoting a more interpretable and generalized representation. The IoU threshold is a hyperparameter that controls the degree of suppression; lower thresholds result in greater diversity but potentially remove valid primitives, while higher thresholds may retain redundant selections.

A Marginal Improvement, Not a Revolution in Physics

Recent advancements demonstrate the capability of Symbolic-KAN, when integrated with Physics-Informed Neural Networks (PINNs), to discern the underlying governing equations of intricate physical systems. This approach moves beyond simply approximating solutions; it actively identifies the mathematical relationships that dictate a phenomenon’s behavior. Successfully applied to equations like the Laplace Equation, which describes electrostatic and gravitational potentials, and the Reaction-Diffusion Equation – crucial for modeling chemical reactions and biological processes – the method reveals a powerful capacity for scientific discovery. By combining symbolic regression with the constraint enforcement of PINNs, the system can effectively ‘learn’ the equations from observed data, offering a novel route to model identification and potentially uncovering previously unknown physical laws. This signifies a shift towards data-driven equation discovery, promising accelerated progress in fields reliant on complex modeling.

The accurate identification of parameters within dynamical systems is crucial for predictive modeling, and this approach exhibits remarkable precision when applied to the Van der Pol Oscillator – a canonical example in nonlinear dynamics. Through rigorous testing, the method consistently estimates oscillator parameters with errors falling below one percent, a significant improvement over existing techniques. This high degree of accuracy isn’t merely a numerical feat; it suggests the system effectively disentangles complex interactions within the oscillator’s behavior, offering a robust foundation for analyzing and predicting similar nonlinear phenomena in fields ranging from electrical engineering to biological systems. The capacity to reliably determine these parameters unlocks the potential for detailed simulations and a deeper understanding of the underlying mechanisms driving the oscillator’s characteristic cycles.

Rigorous testing of Symbolic-KAN against established physics-informed neural networks reveals substantial improvements in accuracy for discerning governing equations. Specifically, when applied to the Reaction-Diffusion Equation, the method demonstrates a remarkable 59% reduction in validation error compared to standard PINN architectures. This performance extends to the Laplace Equation, where Symbolic-KAN achieves an 87% improvement in validation error relative to the cPIKAN method. Beyond overall accuracy, the approach significantly minimizes the magnitude of errors; maximum absolute error is reduced by 70% when contrasted with PINN and by 92% when compared to cPIKAN, indicating a more precise and reliable identification of underlying physical principles from observed data.

The pursuit of interpretable machine learning, as exemplified by Symbolic-KANs, feels less like innovation and more like meticulously reconstructing the wheel. The architecture attempts to embed symbolic structure within a neural network, hoping to extract analytic expressions-a neat trick, if it survives contact with real-world data. It’s a familiar story: elegant theory meets the brutal realities of production. As Marvin Minsky observed, “You can make a case that the brain is a computer that doesn’t know what it’s doing.” This feels painfully accurate. Symbolic-KANs, despite its promise of recovering compact analytic expressions, will likely discover that the universe prefers messy, unexplainable complexity, and any recovered equation is merely a temporary truce with chaos. Tests, predictably, will offer only a fleeting illusion of certainty.

What’s Next?

The promise of recovering analytic expressions directly from data is, predictably, alluring. This work on Symbolic-KANs represents a further attempt to graft interpretability onto deep learning, a pursuit destined to repeat itself in increasingly complex iterations. One suspects the elegance of the recovered expressions will diminish rapidly as problem dimensionality increases, and the symbolic structures will become, if not meaningless, then at least computationally expensive to verify. The true test, of course, will be when these networks encounter data that doesn’t conform neatly to the pre-defined symbolic basis-a scenario production systems invariably locate with startling efficiency.

The architecture’s reliance on a discrete symbolic structure feels… optimistic. It’s a constraint that simplifies the learning task, but at the cost of generality. The next step will inevitably involve relaxing this constraint, allowing for more flexible symbolic representations-and thus, a corresponding increase in the search space. It is a familiar pattern: anything called ‘scalable’ just hasn’t been tested properly. Better one monolith, carefully validated, than a hundred lying microservices claiming to represent fundamental physical laws.

Ultimately, the field will likely gravitate towards hybrid approaches-networks that can seamlessly blend symbolic and numerical reasoning. The real challenge won’t be discovering equations, but determining which equations are actually useful-a question that data alone rarely answers. One anticipates a renewed appreciation for the value of domain expertise-and a healthy skepticism towards any system claiming to automate the scientific method.

Original article: https://arxiv.org/pdf/2603.23854.pdf

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

The Illusion of Control: Why Equations Remain Elusive

Symbolic-KAN: A Patch, Not a Paradigm Shift

Forcing Interpretability: A Game of Gates and Penalties

A Marginal Improvement, Not a Revolution in Physics

What’s Next?

See also: