Unlocking System Dynamics: A New Path to Equation Discovery

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Author: Denis Avetisyan

Researchers have developed a method that combines data-driven learning with inherent physical symmetries to accurately identify the governing equations of complex dynamical systems.

This work introduces a symmetry-informed neural network approach leveraging characteristic curves and symbolic regression for uniquely identifying system equations.

Reconstructing governing equations from data is fundamentally ill-posed, often yielding multiple models that fit observations equally well. This work, ‘Integrating prior knowledge in equation discovery: Interpretable symmetry-informed neural networks and symbolic regression via characteristic curves’, addresses this challenge by extending a characteristic curve (CC)-based framework with both symmetry constraints and post-processing via symbolic regression. The resulting methodology systematically improves the discovery of accurate and interpretable models for dynamical systems, demonstrated on chaotic and discontinuous examples. Can this approach, by effectively integrating prior knowledge, pave the way for more robust and reliable system identification in complex scientific domains?

The Illusion of Accurate Modeling

The ability to accurately model how systems evolve over time is paramount across diverse scientific and engineering disciplines, from predicting weather patterns and managing financial markets to understanding biological processes and designing efficient control systems. However, achieving this fidelity proves remarkably challenging. Traditional reconstruction methods, often reliant on linear approximations or simplified models, frequently falter when confronted with the inherent noise present in real-world data and the complex, nonlinear interactions that govern many dynamic systems. This susceptibility limits their ability to capture subtle yet crucial behaviors, hindering predictive accuracy and potentially leading to flawed interpretations of the underlying mechanisms at play. Consequently, researchers continually seek novel approaches that can robustly handle noisy data and effectively represent the full spectrum of complexity inherent in dynamic systems.

Reconstruction operators, essential tools in system identification, face a fundamental limitation: the amplification of even minor errors during the modeling process. These operators function by translating observed data into a representation of the underlying system dynamics, but each step of this translation-from data acquisition to parameter estimation-introduces potential inaccuracies. Critically, these errors aren’t simply additive; they propagate and accumulate, often exponentially, as the operator iteratively refines the model. Consequently, the resulting identified model may diverge significantly from the true system, particularly in complex or noisy environments. This sensitivity underscores the need for robust reconstruction techniques and careful consideration of error bounds to ensure the fidelity and reliability of identified models, and highlights the ongoing challenge in achieving truly accurate system reconstruction.

Reconstructing a system’s behavior from observed data fundamentally requires solving an inverse problem – deducing the rules that govern the system based on its outputs. This process is inherently challenging because it often involves approximating complex relationships with simplified models, introducing error at each step. Unlike forward prediction, where uncertainties accumulate predictably, inverse problems amplify even minor data imperfections, leading to significant discrepancies between the inferred governing equations and the true system dynamics. This sensitivity stems from the fact that multiple underlying systems can produce similar observed behavior, meaning the solution isn’t unique; any attempt to reverse-engineer the rules relies on assumptions and estimations that inevitably introduce approximation error. Consequently, the fidelity of reconstructed models is often limited, demanding sophisticated techniques to mitigate the propagation of these uncertainties and achieve accurate system representation.

Deconstructing the Chaos: A New Approach

The Neural Network – Characteristic Curves (NN-CC) formalism addresses complex system dynamics by representing them as a composition of individual, single-variable functions – termed characteristic curves. This decomposition allows for the isolation and independent estimation of each contributing factor to the overall system behavior. Instead of modeling the entire system as a monolithic entity, NN-CC breaks it down into modular components, where each characteristic curve describes the relationship between a single input and a specific aspect of the system’s output. This modularity simplifies the modeling process and enables a more focused approach to parameter estimation and system identification. The subsequent composition of these univariate functions then reconstructs the complete system response.

The NN-CC formalism utilizes neural networks to approximate characteristic curves – the fundamental relationships defining a system’s behavior – due to their capacity as universal function approximators. Unlike traditional methods reliant on predefined functional forms, neural networks can efficiently model complex, nonlinear relationships directly from data. This is achieved by training the network to map input variables to the corresponding values on the characteristic curves. The efficiency stems from the network’s ability to learn these mappings with a relatively small number of parameters, particularly when combined with appropriate regularization techniques and network architectures. Consequently, the NN-CC approach provides a data-driven and computationally effective means of representing system dynamics without requiring explicit analytical formulations.

Structural uniqueness within the Neural Network – Characteristic Curves (NN-CC) formalism is achieved by enforcing a specific network architecture and parameterization that maps each input variable to a single, corresponding characteristic curve. This constraint prevents non-uniqueness issues common in standard neural network training, where multiple network configurations can yield the same output. By guaranteeing a one-to-one mapping between inputs and curves, the formalism ensures a well-defined solution for the system dynamics, eliminating ambiguity and increasing the predictability of model outputs. This directly translates to enhanced model reliability, as the solution is not susceptible to variations arising from non-unique parameter sets and reduces the sensitivity to initial conditions during the training process.

The incorporation of symmetry constraints within the Neural Network – Characteristic Curves (NN-CC) formalism significantly enhances both the accuracy and robustness of modeled systems. By explicitly defining and enforcing known symmetries present in the physical system, the NN-CC framework reduces the number of free parameters that require estimation from data. This reduction in parameter space mitigates the risk of overfitting, particularly when dealing with limited or noisy datasets. Consequently, models built with symmetry constraints demonstrate improved generalization capabilities and require less extensive training data to achieve a specified level of performance, effectively improving data efficiency.

Putting Theory to the Test: From Stick-Slip to Duffing

The Neural Network Constrained Component (NN-CC) formalism was validated through application to the Stick-Slip and Duffing oscillators, both established benchmarks for evaluating dynamical systems identification methods due to their non-linear and, in the case of Stick-Slip, non-smooth characteristics. The Stick-Slip oscillator exhibits alternating periods of static friction and kinetic friction, presenting a challenge for traditional methods, while the Duffing oscillator, described by the equation [latex] \ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 = \gamma \cos(\omega t) [/latex], demonstrates complex behaviors including multiple stable states and chaotic motion depending on parameter values. Successful application of the NN-CC formalism to these systems demonstrates its capability to accurately reconstruct governing equations from observed data, even in scenarios with complex dynamic behavior.

To enhance the accuracy of system identification, data smoothing techniques are implemented prior to model extraction. Specifically, the Savitzky-Golay filter-a moving window digital filter-is utilized to reduce high-frequency noise while preserving signal features. This filter operates by fitting a polynomial of a specified degree within the moving window, effectively averaging the data without introducing significant phase distortion. The window size and polynomial order are parameters optimized to balance noise reduction and signal fidelity; larger window sizes provide greater smoothing but may obscure rapid dynamic changes. Application of this filter minimizes the impact of measurement errors and improves the robustness of subsequent analyses, particularly when dealing with noisy experimental data or simulations.

The Neural Network-Constrained Coefficient (NN-CC) formalism demonstrates capability in identifying the governing equations of dynamical systems, even when subject to complexities such as position-dependent friction. This is achieved through the framework’s ability to accurately model frictional forces represented as a second-order Ordinary Differential Equation (ODE). Specifically, the system’s dynamics, including the effects of this [latex] \frac{d^2x}{dt^2} [/latex] friction term, are successfully reconstructed from observed data. Analysis indicates a high degree of fidelity in identifying the coefficients within the governing ODE, validating the formalism’s robustness beyond simple harmonic oscillators and into systems with more complex, non-linear frictional behavior.

Sparse regression and symbolic regression techniques are integrated with the Neural Network Constrained Coordinate (NN-CC) formalism to derive human-interpretable models from complex system dynamics. Sparse regression identifies the most significant terms within a predefined library of potential functions, minimizing model complexity and improving generalization. Symbolic regression, conversely, automatically searches for mathematical expressions that best fit the observed data, without requiring a pre-defined functional form. When applied in conjunction with NN-CC, which provides accurate and noise-reduced state-space coordinates, these regression methods facilitate the discovery of governing equations – including identification of relevant terms and their associated coefficients – that describe the underlying system behavior. This approach allows for the extraction of models expressed in standard mathematical notation, enabling both qualitative understanding and quantitative prediction of system dynamics.

The Illusion of Accuracy: Error and Robustness

Root Mean Squared Error, or RMSE, functions as a crucial benchmark for evaluating the fidelity of reconstructed models to the underlying physical system. This metric quantifies the average magnitude of the error between predicted and actual values, effectively gauging how well the identified model captures the system’s behavior. A lower RMSE indicates a closer alignment between the model’s predictions and the observed data, signifying improved accuracy in representing the dynamics of the system under investigation. By minimizing the [latex]RMSE[/latex], researchers can confidently assess the reliability and predictive power of their identified models, ensuring they accurately reflect the complexities of the physical phenomena being studied and enabling robust simulations and analyses.

Analysis of the stick-slip system reveals a substantial reduction in reconstruction error through the implementation of symmetry constraints. Initial model iterations, without these constraints, yielded a Root Mean Squared Error (RMSE) of approximately 60, indicating a significant discrepancy between predicted and actual system behavior. However, by incorporating known symmetries inherent in the physical system, the RMSE dramatically decreased to around 13. This improvement highlights the power of leveraging prior physical knowledge to guide model learning and achieve greater accuracy in dynamic system reconstruction, demonstrating that incorporating fundamental principles can drastically refine predictive capabilities.

The data-driven modeling approach demonstrated a striking capability – when imbued with sufficient prior knowledge, its performance rivaled that of traditional parametric modeling. This suggests a pathway towards achieving high-fidelity results without relying solely on pre-defined equations or extensive domain expertise. By intelligently integrating physical constraints and leveraging the learning capacity of neural networks, the system effectively bridges the gap between data-driven and physics-based approaches. The resulting accuracy indicates that, given appropriate guidance, machine learning can not only approximate complex dynamics but also provide solutions comparable to those painstakingly derived through analytical methods, opening avenues for efficient and robust system identification in various scientific and engineering domains.

A fundamental principle demonstrated by this work is that minimizing training loss within neural networks isn’t merely about achieving low error on the training dataset, but also about ensuring the network converges towards the actual underlying physical functions governing the system. When the training loss is sufficiently small, the optimization process effectively guides the network’s parameters to a stable state that accurately represents the relationships within the data. This convergence isn’t simply a mathematical property of the optimization algorithm; it reflects a genuine approximation of the true function. Consequently, a sufficiently trained neural network can reliably predict system behavior even for inputs outside of the initial training set, effectively acting as a learned model of the physical process. This suggests that, under appropriate training conditions, neural networks aren’t just ‘black boxes’ but capable of discovering and representing the core principles of a system’s dynamics.

Investigations reveal that a neural network constrained coercion (NN-CC) approach, when augmented with symmetry constraints and followed by post-symbolic regression (SR), demonstrably surpasses the performance of other NN-CC configurations. This optimized methodology not only refines the accuracy of identified models-reducing reconstruction error as measured by Root Mean Squared Error-but also offers a robust framework for extracting interpretable physical laws. The synergy between the neural network’s learning capacity, the guiding influence of symmetry principles, and the analytical power of symbolic regression creates a powerful system capable of uncovering underlying dynamics with increased fidelity and transparency, positioning it as a leading technique in data-driven discovery.

The pursuit of universally applicable models, as demonstrated by this work on characteristic curves and symmetry constraints, feels inevitably transient. The authors aim for structural uniqueness in equation discovery, a noble goal, yet one destined to meet the realities of production data. It echoes a sentiment articulated by Isaac Newton: “If I have seen further it is by standing on the shoulders of giants.” Each refined methodology builds upon previous work, yet each also introduces new failure modes, new edges cases. The elegance of identifying governing equations via CC-based methods will, inevitably, succumb to the noise and complexity of real-world dynamical systems. It’s not a condemnation of the effort, merely an acknowledgment that even the most beautiful abstractions are ultimately temporary footholds in an endless climb.

The Road Ahead

The pursuit of governing equations from data will inevitably encounter the limitations of data itself. This work, with its emphasis on characteristic curves and symmetry, rightly attempts to impose order – a pre-emptive strike against the infinite possibility space. Yet, production data rarely respects elegance. Noise, incomplete observations, and the subtle but persistent drift of real-world systems will test the boundaries of even the most structurally informed algorithms. The promise of ‘unique’ identification feels
 optimistic. It is not that the method is flawed, but that the universe delights in ambiguity.

Future iterations will likely focus on robustness – the ability to gracefully degrade in the face of imperfect data. Expect to see hybrid approaches, integrating this symmetry-informed framework with techniques that explicitly model uncertainty. Furthermore, the computational cost of symbolic regression remains a significant hurdle. A more efficient search space, perhaps guided by learned heuristics, is not merely desirable, but essential for scaling these methods beyond toy problems.

Ultimately, the true test will not be in discovering equations that can describe a system, but in predicting when those equations will fail. A model’s lifespan is finite. Legacy is not a bug; it’s a feature. And in the relentless churn of technological advancement, these beautifully derived equations will, one day, join the vast collection of algorithms that once promised everything, and now simply offer a memory of better times.

Original article: https://arxiv.org/pdf/2601.21720.pdf

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

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2026-01-30 23:27