Physics-Informed Covariates

• Physics-informed covariates are feature representations that encode physical laws and governing equations, enhancing model interpretability and robustness.
• They are constructed using techniques like spectral expansion, dimensional analysis, and kernelization to directly incorporate domain-specific physics into data-driven models.
• Their application improves estimation accuracy, convergence rates, and regularization in tasks such as regression, classification, and field inversion across diverse scientific fields.

Physics-informed covariates are feature representations, transformations, or model components constructed by explicitly encoding physical laws, governing equations, or acquisition parameters relevant to a scientific system. Unlike generic data-driven covariates, these are defined, selected, or transformed according to domain-specific physical mechanisms (e.g., PDE solutions, line integrals, conservation laws, dimensional analysis), with the objective of enhancing interpretability, statistical efficiency, and robustness in regression, classification, surrogate modeling, and data assimilation. Recent research demonstrates that physics-informed covariates yield closed-form estimators, minimax-optimal convergence rates, latent-space compression, and heightened robustness across diverse applications such as spatio-temporal PDE regression, fusion diagnostics, radiative hydrodynamics, medical imaging, support vector machines, Gaussian processes, and field inversion.

1. Mathematical Characterization of Physics-Informed Covariates

Physics-informed covariates are defined by their strict adherence to physical models governing a system. For linear evolution PDEs on a domain $X\times[0,1]$ with operator $\mathcal{L}$, the data,

$U_i = u(X_i, T_i) + \epsilon_i$

are modeled via spectral expansion,

$$u(x, t) = \sum_{k=1}^\infty \alpha_k e^{-\lambda_k t} \psi_k(x)$$

yielding the $K$ physics-informed covariates per sample:

$$Z_{ik} = e^{-\lambda_k T_i} \psi_k(X_i) \qquad (k=1,\ldots,K)$$

These basis-driven features encode time-space decay dictated by the PDE, ensuring that statistical estimation directly respects physical propagation and regularization (Li et al., 2024).

More generally, physics-informed covariates can be constructed via dimensional analysis. For a feature vector $x\in\mathbb{R}^d$ and target $y$ with known units, physics-informed feature maps $\varphi_j(x)$ are dimensionally matched (or are dimensionless) monomials,

$\varphi_j(x) = \prod_{i=1}^d x_i^{a_{ij}}$

where exponents $a_{ij}$ satisfy

$\sum_{i=1}^d a_{ij} D(x_i) = D(y)$

ensuring physical homogeneity and interpretability. This principle recovers known physical laws (e.g., Bernoulli, magnetic dissipation, binding energy) and uncovers new mechanistic relationships in ML applications (Lampani et al., 23 Apr 2025).

2. Construction and Transformation Methodologies

Physics-informed covariate construction is case-dependent and may involve direct encoding, transformation, or kernelization. Prominent approaches include:

• Direct Encoding from Physical Models: Express eigenmode weights, decay rates, and solution operators as feature maps. For example, in MRI segmentation, acquisition parameters TR, TE, flip angle and their transforms ($e^{-TR}$, $\sin\theta$) are provided to a neural network as auxiliary inputs alongside voxel data (Borges et al., 2020).
• Physics-Informed Transformation of Variables: In radiative hydrodynamics ML surrogates, absorption ($\alpha_g$) and emissivity ($\epsilon_g$) are transformed to net energy exchange ($\delta_g$) and scaled photon mean free path ($\tau_g$):

$$\delta_g = e_g - a_g = 4\pi (\Delta E)_g \epsilon_g / h - c \alpha_g U_g$$

$$\tau_g = \sigma\left(\frac{\ln l_g - \frac{1}{2} \ln(l_\mathrm{max} l_\mathrm{min})}{\ln(l_\mathrm{max}/l_\mathrm{min})}\right),\quad l_g = 1/\alpha_g$$

These transforms compress crucial physics (diffusion, energy coupling) and improve dimension reduction and surrogate fidelity, outperforming log and root transforms by orders of magnitude (Cho et al., 2024).

• Physics-Informed Kernelization: In support vector machines for high energy physics, the kernel is designed to mirror underlying cross-section analytics:

$$\kappa_\text{phys}(x, z) = \gamma(\langle x, z\rangle^2 + \langle x, z\rangle + \langle x, z\rangle \exp(\langle x, z\rangle))$$

where $x, z$ are vectors of kinematic observables (e.g., $E$, $p_T$, rapidity $y$, angle $\phi$), and $\gamma$ is a scaling parameter. This reflects resonant structure and conservation laws essential for collider signal-background discrimination (Ramirez-Morales et al., 2024).

• Physics-Informed Covariance in GPR: Physics-Information-Aided Kriging constructs mean/covariance directly from stochastic PDE simulation outputs, enforcing physical constraints and resulting in non-stationary, physically consistent GP priors (Yang et al., 2018).
• Line-Integral Encoding for Diagnostic Inversion: In fusion diagnostics, chord geometry is encoded as a tensor $P$ (stacked contribution matrices $C_i$) and fused into deep models at the feature level, with corresponding loss terms enforcing line-integral consistency, thus directly integrating measurement geometry into the reconstruction (Wang et al., 2024).

3. Statistical Properties and Computational Considerations

Physics-informed covariates confer both statistical and computational advantages due to their low intrinsic dimension, interpretability, and natural regularization:

• Closed-form Estimators: Truncated modal expansion enables ordinary least squares estimation without additional penalty, regularizing the inverse problem via explicit eigenmode selection (Li et al., 2024).
• Optimal Convergence Rates: For estimators of the initial profile $g_0$ in parabolic PDE regression, bias-variance trade-off yields rates

$\|\hat{g} - g_0\|^2 = O_P(n^{-(2s-1)/(r+2s)})$

with a matching minimax lower bound (Li et al., 2024).

• Latent-Space Compression: Physics-informed transformation (e.g., $\delta_g$, $\tau_g$) in ICF surrogate models reduces necessary principal component dimensions by factors of 10–100 for fixed physical error thresholds (Cho et al., 2024).
• Enhanced Regularization and Data Efficiency: Physics-informed field inversion appends a spatially dense PDE residual to the loss, acting as a spatially adaptive regularizer. This enables accurate field recovery under limited, truncated, or noisy observations, outperforming purely data-driven inversion at modest computational overhead (Ugur et al., 23 Sep 2025).
• Algorithmic Complexity: In modal-based regression, the cost is $O(nK + K^3)$, and K can be selected by cross-validation or BIC. For GP-based workflows, MLMC further reduces sampling cost by leveraging coarse/fine solver hierarchies (Yang et al., 2018).

4. Domain-Specific Applications

Physics-informed covariates are foundational across multiple scientific domains:

• Spatio-Temporal Dynamical Systems: Modal-based regression for PDE-constrained dynamics enables principled interpretability and parsimonious computation in the analysis of diffusion, wave propagation, and multi-field systems (Li et al., 2024).
• Inertial Confinement Fusion Simulation: Physically transformed NLTE model inputs (energy exchange, transport lengths) align surrogates with simulation drivers, resulting in improved dimensionality reduction and reduced physical error (Cho et al., 2024).
• Brain MRI Segmentation: Inclusion of scan acquisition parameters as physics-informed covariates makes CNN-based segmentation robust to protocol variability, enabling harmonized multi-centre and longitudinal studies (Borges et al., 2020).
• High-Energy Physics Classification: Physics-informed kernels in SVM classifiers for processes such as Drell–Yan Z boson production yield superior classification metrics (AUC, accuracy, precision) under severe class imbalance relative to vanilla kernel methods (Ramirez-Morales et al., 2024).
• Field Inversion and Assimilation: Incorporation of PDE loss terms into field inversion leads to enhanced recovery accuracy for spatially varying parameters in turbulence and heat conduction even with sparsely observed data (Ugur et al., 23 Sep 2025).
• Line-Integral Diagnostics in Fusion: Deep architectures (e.g., Onion) utilizing measurement geometry tensors and physics-informed loss achieve substantial reduction in reconstruction error and improved surrogate generalization across fusion devices (Wang et al., 2024).

5. Error Metrics and Physical Interpretability

Physics-informed covariates enable the definition, direct use, and minimization of physically relevant error metrics:

• Groupwise Physical Errors: In ML surrogates for NLTE, errors are measured in physically transformed spaces,

$$L1_\delta = \sum_g |\delta_g - \delta_g^\text{apx}|,\quad L2_\delta = \sum_g (\delta_g - \delta_g^\text{apx})^2$$

preserving groupwise mismatches and correlating directly to simulation drivers, in contrast to conventional scalar metrics (Planck/Rosseland means, integrated emissivity) (Cho et al., 2024).

• Physical Covariance Preservation: In physics-informed Kriging, linear constraints from SPDEs are exactly preserved in the GP posterior mean, and physically meaningful uncertainty quantification is obtained (Yang et al., 2018).
• Adjoint Gradient Information: Physics-informed residual loss terms in field inversion supply dense gradient information, regularizing parameter corrections and reducing overfitting to noise (Ugur et al., 23 Sep 2025).
• Loss Terms Enforcing Measurement Consistency: In line-integral inversion models, the loss

$$\text{loss}_2 = \frac{1}{M} \sum_j \|C\,\text{Net}(x_j) - x_j\|_2^2$$

brings model output into compliance with real measurement physics, reducing inversion error and profile mismatch (Wang et al., 2024).

6. Experimental Demonstrations and Quantitative Impact

Across applications, physics-informed covariates systematically outperform generic counterparts:

Domain	Approach	Metric	Generic Covariate	Physics-Informed Covariate
Spatio-temporal regression	Modal OLS	$\|\hat{g}-g_0\|^2$ rate	Suboptimal	Minimax optimal
ICF NLTE surrogate	PCA dimension vs. error	L2 error ($\tau_g$)	$>10^{-1}$	$<10^{-2}$
MRI segmentation	Dice score (GM, WM)	Across protocol	0.904, 0.943	0.910, 0.948
HEP SVM	AUC (1:10 imbalance)		0.78 (RBF)	0.96 ($\kappa_\text{phys}$)
Kriging/Field inversion	$L^2$ error, R²	Sparse/noisy data	$>40\%$	$14\%$ (MLMC+PhIK)
Line-integral inversion	Relative error (E₂, phantom)		$4.88\times 10^{-2}$	$0.83\times 10^{-2}$

These results consistently support improved robustness, error reduction, and interpretability when physical laws are hardwired into covariate construction.

7. Generalization and Extensions

Physics-informed covariates and design principles generalize to:

• Time-dependent and nonlinear operators: Joint time–space spectral bases, split-step diagonalization, hyper-reduction, and discrete empirical interpolation for nonlinear PDEs (Li et al., 2024).
• Multi-field and multi-physics systems: Modal stacks of matrix-valued eigenmodes, cross-covariances, and composite kernelization to encode coupled dynamics (Li et al., 2024).
• Active learning and experimental design: Variance maximization strategies in physics-informed Kriging efficiently guide new measurements to regions of model uncertainty (Yang et al., 2018).
• Explainable machine learning: Automated feature ranking among physics-informed covariates identifies dominant mechanisms, with potential for discovery of previously unrecognized conserved quantities (e.g., magnetic helicity in solar flares) (Lampani et al., 23 Apr 2025).
• Hybrid data-physics inversion: Physics-informed field inversion leverages sparse data and adaptive physical regularization, extending applicability to regimes where generic methods fail (Ugur et al., 23 Sep 2025).

Physics-informed covariates thus represent a unifying principle for model-based statistical learning in scientific applications, yielding interpretability, efficiency, error guarantees, and adaptability across domains.