Physics-Guided Inference

• Physics-guided inference is a methodology that integrates physical laws, priors, or constraints into statistical estimation to enhance model parameter recovery and prediction accuracy.
• It leverages domain-specific features like symmetries, conservation laws, and stochastic fluctuations to inform efficient and precise inference strategies.
• Tuning environmental parameters such as temperature enables active learning protocols that maximize Fisher information and optimize model reconstruction.

Physics-guided inference refers to methodological strategies for integrating physical laws, priors, or constraints directly into the inference process—statistical estimation, learning, or model identification—of systems governed by underlying physical dynamics. This paradigm leverages domain-specific properties (such as symmetries, conservation laws, or stochastic fluctuations) to optimize information extraction, model parameter identification, and prediction in natural and engineered systems where data arise from the stochastic or deterministic evolution of a physical process. Physics-guided inference contrasts with purely data-driven learning by treating the environment’s physical characteristics and control parameters as central to the informativeness and efficiency of statistical estimates.

1. Information-Theoretic Foundations

Physics-guided inference in physical systems, as formalized for spin-network inference (Huang et al., 2018), is grounded in information theory. The central construct is the Fisher information matrix I, which quantifies the curvature of the log-likelihood landscape concerning model parameters and therefore sets local statistical distinguishability and achievable inference accuracy. For equilibrium data sampled from a Boltzmann distribution

$P(s|J) = \frac{\exp[-E(s|J)/T]}{Z},$

with energy

$E(s|J) = -\sum_{(i,j)} J_{ij} \sigma_i \sigma_j,$

the Fisher information for each parameter $J_{ij}$ is

$$I_{(ij),(ij)} = \frac{\operatorname{Var}[\sigma_i \sigma_j]}{T^2},$$

emphasizing that the informativeness of data about physical couplings is intimately coupled to the magnitude and structure of statistical fluctuations in the observed configurations, as modulated by the temperature $T$ (environmental noise level).

The Kullback–Leibler divergence $D[P(s|J)||P(s|J')]$ measures the statistical separation between models and, for small $|J-J'|$, is determined by the local Fisher information. Thus, the formulation links statistical distinguishability, observable variances, and control parameters such as temperature—providing a quantitative basis for optimizing inference protocols subject to physical constraints.

2. Dependence on Physical Properties and Environmental Parameters

The nature of the physical system and its environment crucially determines both the amount and quality of information available for inference. In the canonical spin-network example, the observable statistics at low temperatures ($T \rightarrow 0$) are dominated by energetically favored ground states, leading to minimal exploration of the state space and severe degeneracy: many network interaction parameters $J$ yield indistinguishable observational distributions. At high $T$, thermal agitation produces near-uniform sampling over all $2^m$ possible states, erasing distinctions among models. The informative regime emerges from an interplay between these extremes: only stochastic dynamics at appropriate $T$ lead to sufficiently broad but not overwhelmingly noisy data, maximizing the relevant variances for Fisher information and minimizing local uncertainty.

Physical system topology, interaction strengths, and environmental control variables (such as magnetic fields) further define the geometry of the likelihood landscape and, thus, the achievable precision. For instance, physical features like the magnitude and fluctuation of spin–spin correlations directly control $I_{(ij),(ij)}$, with

$$\operatorname{Var}[\sigma_i \sigma_j] = 1 - \langle \sigma_i \sigma_j \rangle^2$$

for zero-mean spins, linking observable moments to the Fisher information available for model identification.

3. The Role of Stochastic Fluctuations

Stochastic fluctuations, governed by the system’s temperature or analogous noise parameters, play a dual role in physics-guided inference. On one hand, they inject variability, driving the system away from a limited set of ground states and generating a representative set of configurations over which differences in the energy landscape (and thus in the parameters $J$) become observable and discriminable. On the other, excessive fluctuations smooth over energetic differences, reducing the sensitivity of system statistics to variations in $J$, as reflected in the inverse quadratic dependence of $I_{(ij),(ij)}$ on temperature.

Thus, there exists a non-trivial trade-off: fluctuations are essential to probe the state space but must be regulated (modulated by $T$ or analogous parameters). The inference task is most efficient when this balance produces maximal statistical curvature, i.e., the largest diagonal Fisher information, for the system’s relevant parameters.

4. Optimal Physical Regimes for Inference

The existence of an intrinsic “optimal” temperature $T_{\text{opt}}$ is a key insight of physics-guided inference. This optimal point maximizes the Fisher information matrix, corresponding to the least ambiguous statistical distinctions between competing parameterizations. Empirically, $T_{\text{opt}}$ aligns with physically significant points such as phase transitions or critical phenomena in the network. As established via explicit calculation and graphical analysis (Huang et al., 2018), the Fisher information increases monotonically from low $T$, peaks at $T_{\text{opt}}$, and decreases thereafter. This principle generalizes to more complex systems, suggesting that optimal physical regimes for statistical learning coincide with maximal system variability short of global disorder—connected to criticality and the emergence of rich fluctuations.

5. Protocols for Physics-Guided Active Learning

Physics-guided inference enables strategic, active learning protocols that exploit the tunability of environmental parameters. The proposed protocol proceeds in two stages:

1. Parameter sweep: The observer empirically estimates the Fisher information across a range of temperatures (or other physical controls), identifying the temperature $T_{\text{opt}}$ that maximizes the minimal diagonal element of the Fisher information matrix:

$$T_{\text{opt}} = \operatorname{argmax}_T \left[\min_{(i,j)} I_{(ij),(ij)}(J, T)\right]$$

1. Focused inference: Additional equilibrium data are collected at $T_{\text{opt}}$, and standard maximum-likelihood estimation—often via gradient ascent—is used to reconstruct the underlying parameters $J$.

This protocol has been experimentally validated, for example, in neural circuit reconstruction tasks, where driving the system to $T \approx 11$ produced lower $L_2$ error in the recovered network topology compared to observations at $T = 1$. The protocol demonstrates that physical adaptation—tuning $T$—substantially enhances data efficiency and inference fidelity.

6. Synthesis: The Physics–Information Link and Generalization

The fundamental conclusion of the physics-guided inference framework is that the sample complexity and efficiency of inference in systems governed by physical laws are not solely determined by abstract model class complexity, but are constrained and modulated by the dynamics and stochasticity of the data-generating system. This has broad implications across natural sciences: in neural circuits, gene regulatory systems, or economic agents, the informativeness and reliability of inferences can be optimized by tuning the physical environment to achieve the “sweet spot” between variability and signal clarity.

Physics-guided inference thus offers a unified methodology: integrating information-theoretic quantitative analysis, physical control of environmental variability, and modern machine learning techniques—culminating in protocols that systematically maximize inference efficiency. This perspective extends naturally to more complex systems and may inform the design of experiments, control strategies, and computational learning approaches for a wide range of physical, biological, and engineered networks.