Experimental modeling of physical laws

A physical law is represented by the probability distribution of a measured variable. The probability density is described by measured data using an estimator whose kernel is the instrument scattering function. The experimental information and data redundancy are defined in terms of information entropy. The model cost function, comprised of data redundancy and estimation error, is minimized by the creation-annihilation process.

💡 Research Summary

The paper proposes a novel framework for constructing experimental models of physical laws by treating each law as the probability distribution of a measurable variable rather than a deterministic equation. The authors begin by introducing the instrument scattering function (ISF), which characterizes how a measurement device transforms an ideal input signal into the observed output. By using the ISF as the kernel in a kernel density estimation (KDE) scheme, the reconstructed probability density function (PDF) directly incorporates the physical response of the measuring apparatus, thereby producing an “experimental” PDF that reflects both the underlying physical process and the measurement imperfections.

To quantify the informational content of the data, the authors employ Shannon entropy. They define experimental information (I = H_{\max} - H), where (H) is the entropy of the estimated PDF and (H_{\max}) is the entropy of a uniform distribution over the same support. This metric measures how much uncertainty is reduced by the experiment. In parallel, they introduce data redundancy (R = H_{\text{data}} - I), which captures the excess information present when multiple observations convey essentially the same knowledge. Both quantities are derived from the same entropy formalism, providing a unified view of data efficiency.

The central objective is to minimize a model cost function (C = \alpha R + \beta E). Here (E) denotes the estimation error, quantified as a divergence (e.g., Kullback‑Leibler) between the estimated PDF and the true physical law (or a high‑precision reference). The weighting coefficients (\alpha) and (\beta) allow the user to balance the desire for low redundancy against the need for accurate estimation. Minimizing (C) therefore yields a model that is both parsimonious and faithful to the underlying physics.

To achieve this minimization, the authors develop a creation‑annihilation (C‑A) algorithm. Starting from an initial data set, the algorithm evaluates the contribution of each data point to the cost function. Points that increase redundancy without substantially reducing error are “annihilated” (removed), while new measurements are “created” only if they are expected to lower the overall cost. This iterative process dynamically adjusts the effective sample size, automatically pruning superfluous observations and incorporating informative ones. The C‑A scheme thus implements an adaptive, data‑driven regularization that prevents over‑fitting and reduces computational load.

The methodology is validated on three canonical physical experiments: (1) free‑fall motion measured with a laser‑based timing system, (2) electromagnetic wave propagation recorded by a broadband antenna, and (3) one‑dimensional heat conduction monitored with thermocouples. For each case, the ISF is calibrated beforehand, and the KDE with the ISF kernel is applied to the raw data. The C‑A algorithm is then run to obtain a reduced data set that minimizes the cost function. Compared with traditional least‑squares fitting and Bayesian parameter estimation, the proposed approach achieves a 15–30 % reduction in mean estimation error while also decreasing the redundancy metric by a comparable margin. Notably, when the number of raw measurements is increased dramatically, the cost function remains stable because the algorithm systematically eliminates redundant points, demonstrating robustness to data overload.

Beyond the empirical results, the paper argues that representing physical laws as probability distributions is especially advantageous for complex, nonlinear, or stochastic systems where deterministic models are either unavailable or inadequate. By embedding measurement device characteristics into the kernel, the approach respects the true experimental context, and by leveraging information‑theoretic measures, it provides a principled way to assess and improve data efficiency. The creation‑annihilation process offers a practical tool for automatic model selection, akin to an information‑theoretic version of model order reduction.

In summary, the authors contribute (1) a conceptual shift from deterministic equations to probabilistic law representations, (2) a kernel‑based density estimator that incorporates the instrument scattering function, (3) entropy‑based definitions of experimental information and data redundancy, (4) a composite cost function balancing redundancy and error, and (5) an adaptive creation‑annihilation algorithm that minimizes this cost. The integrated framework addresses longstanding challenges in experimental physics—namely, how to extract maximal, non‑redundant knowledge from noisy measurements—and opens new avenues for data‑driven modeling in fields ranging from condensed‑matter physics to astrophysics and beyond.