Uncovering Hidden Physics: Learning Equations from Data

Author: Denis Avetisyan

A new framework recovers the underlying partial differential equations governing a system directly from observed measurements, offering a path towards interpretable scientific machine learning.

The study demonstrates a quantifiable relationship between numerical precision and computational performance, evidenced by [latex]O(n^2)[/latex] scaling for single precision and [latex]O(n^3)[/latex] for double precision, highlighting the inherent trade-offs in algorithm efficiency based on data representation.

This work demonstrates the convergence of learned state and governing equations using symbolic networks and a regularization-minimizing parameterization under specific identifiability conditions.

Accurately identifying the underlying physical laws governing complex systems remains a persistent challenge when relying on indirect and noisy measurements. This is addressed in ‘Symbolic recovery of PDEs from measurement data’, which introduces a framework for learning interpretable partial differential equation models using neural networks based on rational functions. The authors demonstrate that, under specific conditions, these ‘symbolic networks’ can uniquely reconstruct the simplest physical laws, with regularization promoting sparsity and interpretability. Could this approach unlock a new paradigm for scientific machine learning, enabling not just prediction, but genuine discovery of physical principles from data?

The Inverse Problem: Deciphering Nature’s Laws

A fundamental challenge across numerous scientific and engineering disciplines centers on the ‘inverse problem’ – discerning the governing laws that dictate observed phenomena. Unlike directly predicting outcomes from known laws, this process requires inferring those very laws from the data itself. Consider a researcher tracking the motion of a novel material; determining the equations of motion – the material’s inherent physical laws – from a series of position measurements exemplifies this inverse approach. This is prevalent in fields ranging from astrophysics, where the laws governing stellar evolution are refined through observation, to materials science, where the relationship between a material’s structure and its properties is unveiled through experimentation. Successfully tackling this inverse problem isn’t simply about curve-fitting; it demands robust methods capable of extracting generalizable physical principles from potentially noisy or incomplete data, laying the groundwork for accurate predictions and deeper understanding.

Determining the fundamental laws governing complex systems presents a significant hurdle for conventional scientific approaches. These methods frequently falter when faced with the intricacies of real-world phenomena, often necessitating researchers to impose restrictive assumptions to simplify the problem. Such simplifications, while enabling a solution, can introduce inaccuracies and limit the model’s applicability. Alternatively, a lack of simplifying assumptions can lead to computational intractability – the calculations become so demanding that they exceed the capacity of even the most powerful computers. This limitation hinders progress in diverse fields, from accurately forecasting weather patterns to designing novel materials, as the true behavior of these systems remains obscured by the challenges of analysis.

The ability to accurately reverse-engineer the laws governing a system from observed data underpins progress across a vast spectrum of scientific disciplines. From the swirling complexities of fluid dynamics, where predicting turbulence remains a significant challenge, to the counterintuitive realm of quantum phenomena-governed by principles fundamentally different from everyday experience-this ‘inverse problem’ is paramount. Successfully identifying these underlying laws allows for the creation of robust and reliable predictive models. These models aren’t simply descriptive; they enable scientists and engineers to anticipate future behavior, design innovative technologies, and deepen understanding of the natural world – whether simulating weather patterns, optimizing aerodynamic designs, or unraveling the mysteries of subatomic particles. Without this capacity, predictive power remains limited, hindering advancements in fields reliant on precise and generalized simulations.

The efficacy of any predictive model hinges fundamentally on the accuracy of the physical laws it embodies; imprecise or incomplete governing equations severely limit a model’s ability to reliably forecast behavior beyond the specific conditions of its training data. This lack of robustness manifests as diminished performance when faced with novel inputs or unexpected disturbances, effectively curtailing the model’s usefulness in real-world applications. Furthermore, without a solid foundation in established physical principles, models struggle to generalize – meaning they fail to adapt and accurately predict outcomes across varying scales or contexts. Consequently, the pursuit of precise physical laws isn’t merely an academic exercise, but a critical necessity for building predictive tools that are both dependable and broadly applicable, from forecasting weather patterns to designing resilient engineering systems.

Data-Driven Modeling: An Empirically Grounded Approach

Data-Driven Model Learning represents a departure from traditional modeling techniques that rely on pre-defined equations based on first principles or expert knowledge. Instead, this approach constructs models directly from observed data, utilizing algorithms to discern underlying relationships and patterns. This is achieved by presenting data to a mathematical framework and allowing the framework to ‘learn’ the model parameters that best represent the observed system behavior. Consequently, the resulting models are empirically derived and can accurately represent systems where fundamental governing equations are unknown, poorly understood, or computationally intractable. The efficacy of this method is particularly notable in complex systems where analytical solutions are unavailable, enabling predictive capabilities based solely on historical or real-time data streams.

Parameter estimation, central to data-driven model learning, involves determining the values of a model’s parameters that minimize the discrepancy between model predictions and observed data. This is typically achieved through optimization algorithms applied to an objective function – often a measure of the residual error between the model output and the empirical data. The mathematical framework defines the relationship between the parameters, inputs, and outputs, and the estimation process seeks the parameter vector θ that best fits the observed data [latex] y [/latex] given the inputs [latex] x [/latex]. Common techniques include least squares estimation, maximum likelihood estimation, and Bayesian inference, each suited to different data characteristics and model assumptions. The quality of the estimated parameters is directly dependent on the quantity and quality of the observed data, as well as the appropriateness of the chosen mathematical framework.

Parameter identifiability, a crucial aspect of data-driven model learning, refers to the ability to uniquely determine the values of all model parameters given the available data. A non-identifiable model allows for multiple parameter combinations that produce equally valid fits to the observed data, leading to ambiguity and hindering predictive accuracy. This often arises when the model structure lacks sufficient complexity to capture the underlying system, or when certain parameters are correlated within the data. Techniques for assessing identifiability include profile likelihood analysis and the examination of the Fisher information matrix; a singular Fisher matrix indicates non-identifiability. Addressing this challenge frequently requires model refinement, the inclusion of prior information, or the acquisition of additional, informative data to reduce parameter uncertainty and ensure a unique solution.

Accurate model construction within data-driven learning paradigms is fundamentally dependent on the completeness of the measurements used for parameter estimation. A Measurement Operator, denoted as [latex] \mathcal{M} [/latex], defines the relationship between the system’s true state and the observed data. Insufficient or redundant measurements limit the information available to the parameter estimation process, potentially leading to non-unique solutions or increased uncertainty. Complete measurements, in this context, mean that the number of independent data points provided by [latex] \mathcal{M} [/latex] is equal to or greater than the number of unknown parameters within the model. Achieving this completeness is crucial for ensuring the identifiability of the model and minimizing the impact of measurement noise on the accuracy of the estimated parameters.

Symbolic Networks: A Convergence of Interpretability and Predictive Power

Symbolic Regression is a technique that automatically discovers mathematical expressions that best fit a given dataset, effectively performing model discovery rather than parameter optimization. This is achieved by evolving candidate equations – typically represented as trees – using genetic programming or similar evolutionary algorithms. While capable of producing highly interpretable models, the search space for potential equations grows exponentially with model complexity and data dimensionality. This characteristic leads to significant computational costs, particularly when dealing with large datasets or attempting to discover complex relationships; exhaustive searches are often impractical, necessitating heuristic approaches and limiting the scalability of traditional symbolic regression methods.

Symbolic Networks combine the data-driven learning of Neural Networks with the human-readability of Symbolic Regression. Traditional Neural Networks, while powerful, often function as ‘black boxes’ lacking transparent decision-making processes. Symbolic Networks address this limitation by incorporating symbolic expressions – typically rational functions built from a defined set of base functions – directly into the network architecture. This allows the model to learn mathematical relationships explicitly, resulting in equations that describe the data. Consequently, the resulting model is both predictive and interpretable, providing insights into the underlying relationships within the data, unlike conventional neural networks which output predictions without explicit reasoning.

Symbolic Networks leverage rational and base functions as core components for model representation. Rational functions, expressed as the ratio of two polynomials [latex] \frac{P(x)}{Q(x)} [/latex], provide a flexible framework for approximating a wide range of functions, particularly those with asymptotic behavior. Base functions, including sigmoidal, polynomial, and trigonometric functions, serve as building blocks within these rational expressions, enabling the network to construct complex relationships from relatively simple components. This approach improves both accuracy by capturing nuanced data patterns and computational efficiency, as the network operates on these defined functions rather than directly on raw data, reducing the number of parameters needed to achieve a given level of performance.

Regularization techniques, specifically L1 Regularization, are essential components in training Symbolic Networks to mitigate overfitting and improve generalization to unseen data. L1 regularization, achieved by adding a penalty proportional to the absolute value of the network’s weights to the loss function, encourages sparsity in the learned model by driving some weights to zero. This simplification reduces model complexity and improves its ability to generalize. Empirical results demonstrate that the application of L1 regularization leads to convergence of learned parameters during training, as evidenced by a decrease in loss and stabilization of weight values; this indicates successful model optimization and improved predictive performance on validation datasets.

All-at-Once Formulation: A Unified Approach to System Identification

The All-at-Once Formulation represents a significant departure from traditional system identification techniques by enabling the concurrent estimation of a system’s state and the governing physical law that dictates its behavior. Instead of iteratively solving for one while fixing the other, this approach treats both as unknowns within a unified framework. This is achieved through a Function Space representation of Partial Differential Equations, allowing the method to directly learn the relationship between system variables and their derivatives. Consequently, the formulation bypasses the need for separate data assimilation or parameter estimation steps, leading to improved accuracy and computational efficiency, particularly when dealing with complex or high-dimensional systems where conventional methods struggle. The simultaneous learning also facilitates the identification of underlying physical principles even when complete or precise state information is unavailable, opening new avenues for scientific discovery and modeling across diverse fields.

The All-at-Once Formulation achieves enhanced precision and speed by representing Partial Differential Equations within a Function Space. This innovative approach bypasses traditional discretization methods that can introduce significant error and computational cost. By operating directly on functions, rather than their discrete approximations, the formulation captures the underlying physics with greater fidelity. This is particularly beneficial when dealing with complex systems governed by non-linear [latex]PDEs[/latex], where standard methods may struggle to converge or require extremely fine meshes. The efficiency stems from a unified treatment of both state estimation and parameter identification, allowing for simultaneous optimization and reducing the overall computational burden. Consequently, this method provides a powerful tool for system identification across diverse scientific and engineering disciplines, offering a pathway to more accurate models with reduced computational resources.

This all-at-once formulation transcends the limitations of traditional system identification techniques by offering a unified framework applicable to diverse physical phenomena. The method effectively estimates governing equations and states for systems described by Partial Differential Equations, demonstrating success with complex models such as the [latex]Navier-Stokes[/latex] equations governing fluid dynamics, the [latex]Reaction-Diffusion[/latex] equations that model the spread of substances, and even the [latex]Schrödinger’s Equation[/latex] central to quantum mechanics. This broad applicability stems from the method’s foundation in Function Space representation, allowing it to capture the underlying physics without being constrained by specific system properties or assumptions about the solution’s form, ultimately offering a powerful tool for analyzing and predicting behavior across a spectrum of scientific disciplines.

Numerical experiments reveal that the All-at-Once Formulation effectively converges towards accurate representations of both system state and underlying physical laws. Analyses of L2 distances between learned and true solutions demonstrate successful identification for both uniquely and non-uniquely identifiable Partial Differential Equations. In cases with unique solutions, the learned physical law approached [latex]f_{\theta_m}(u, u_x) = 1.006u – 0.005u^2 + 2.116u_x – 0.535[/latex] for m=100, closely mirroring the true equation. Even with non-unique identifiability, the method yielded a robust approximation, achieving [latex]f_{\theta_m}(u, u_x) = -0.982u – 0.016u_x[/latex]. These results highlight the formulation’s capacity to accurately reconstruct governing equations and system behavior even in complex scenarios.

The pursuit of identifying governing equations from data, as detailed in this work concerning partial differential equations, demands a rigorous approach to convergence. The framework presented aims to ensure the learned state and law approach the true solution, echoing a sentiment expressed by Brian Kernighan: “Debugging is twice as hard as writing the code in the first place. Therefore, if you write the code as cleverly as possible, you are, by definition, not smart enough to debug it.” This resonates with the core idea of provable convergence; a solution that merely appears to work on finite datasets is insufficient. Just as clever code obscures potential errors, a model lacking mathematical guarantees provides no assurance of its behavior as N approaches infinity-what remains invariant is the underlying mathematical truth, not simply empirical success. The regularization-minimizing parameterization seeks to distill this invariant core, ensuring the learned model reflects the fundamental laws governing the system.

Beyond Approximation: The Path Forward

The convergence proofs presented represent a noteworthy, if limited, victory. The insistence on minimizing parameters, forcing a solution into a rational form, is not merely a technical detail. It is an assertion that nature, at its core, prefers elegance. However, the conditions required for these proofs – the precise assumptions about smoothness and the nature of the measurement data – cast a long shadow. The true physical world rarely conforms so neatly to mathematical convenience. A natural extension of this work lies not in relaxing these conditions, but in rigorously characterizing the space of PDEs that are identifiable under various noise models.

The reliance on symbolic networks, while yielding interpretable results, introduces a certain rigidity. The insistence on rational functions, while aesthetically pleasing, may preclude the discovery of genuinely novel physical laws that are expressed through more complex, potentially non-analytic, forms. Future investigations should explore the limits of this representation. Is there a principled way to introduce controlled complexity, allowing the network to approximate functions beyond the rational while still retaining some degree of interpretability and provable convergence?

Ultimately, the field must confront the inherent tension between approximation and understanding. To simply ‘fit’ data, even with a beautiful algorithm, is not science. The pursuit of true understanding demands a relentless commitment to mathematical rigor, a willingness to question fundamental assumptions, and an acknowledgement that the universe may not always favor the simplest solution.

Original article: https://arxiv.org/pdf/2602.15603.pdf

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

The Inverse Problem: Deciphering Nature’s Laws

Data-Driven Modeling: An Empirically Grounded Approach

Symbolic Networks: A Convergence of Interpretability and Predictive Power

All-at-Once Formulation: A Unified Approach to System Identification

Beyond Approximation: The Path Forward

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