Physics-Informed Uncertainty (PIU)

• Physics-Informed Uncertainty (PIU) is an approach that integrates physical law constraints, such as PDE/ODE residuals, into quantifying predictive uncertainty in machine learning models.
• It employs methodologies like residual-based penalties, physics-informed priors, and conformal prediction to calibrate uncertainty estimates in both parametric and nonparametric frameworks.
• PIU enhances model reliability by providing calibrated intervals and error bounds, ensuring statistical consistency and out-of-distribution awareness.

Physics-Informed Uncertainty (PIU) denotes a rigorous family of uncertainty quantification (UQ) strategies in which physical laws, typically expressed as differential equation constraints, are exploited to regularize, inform, or define the structure of predictive uncertainty in scientific machine learning. PIU unifies several approaches: (i) constraining predictive distributions via PDE/ODE residuals within parametric or nonparametric models, (ii) leveraging known physics to generate residual-based uncertainty scores or error bounds, (iii) calibrating surrogate models using physics-consistent priors or discrepancy processes, and (iv) integrating conformal or fiducial inference procedures directly in the residual or physics space. The overarching goal is to enable statistical reliability, calibrated intervals, and out-of-distribution awareness in data-driven models, while simultaneously respecting physical law.

1. Mathematical Principles of Physics-Informed Uncertainty

Physics-Informed Uncertainty operationalizes UQ in machine learning models by embedding physical constraints into the learning and calibration of uncertainty representations. The dominant mathematical principle is to enforce or penalize the residuals of governing equations (e.g., $\mathcal{L}u = b$), so that the uncertainty associated with predicted physical quantities reflects both data/model mismatch and the extent to which predictions obey underlying physics.

Key paradigms include:

• Residual-based penalty or constraint: Introducing terms such as $$\mathbb{E}_{x,z}\|\mathcal{L}g_\theta(x,z)-b(x)\|^2$$ in loss functions, as in physics-informed neural networks (PINNs) and physics-informed GANs (Gao et al., 2021).
• Physics-informed priors and kernels: Defining GP priors or covariance structures so that differential operators act linearly on the process, enabling multi-output GPs that encode both the state and its physics-induced derivatives (Spitieris et al., 2022).
• Conformal prediction using residuals: Quantifying uncertainty via calibrating the distribution of residual norms from the PDE operator, yielding intervals or bands with finite-sample coverage in the residual (not data) space (Gopakumar et al., 6 Feb 2025).
• Physics-based discrepancy modeling: Introducing explicit discrepancy functions (e.g., GP terms) to account for model-form uncertainty when physical laws are approximate (Spitieris et al., 2022).
• Fiducial and hybrid inference schemes: EFI-based approaches construct data-driven confidence sets by treating physics-constrained mappings as structural equations, learning the joint law of model parameters and unmodeled errors (Shih et al., 25 May 2025).

Across these variants, the quantification and propagation of uncertainty—aleatoric, epistemic, and model-form—are fundamentally tied to the structure, or violation, of the underlying physics.

2. Representative Frameworks and Methodologies

PIU encompasses a suite of specific algorithmic frameworks, each tailored to particular problem classes and uncertainty regimes:

Framework / Method	Physics Role	Uncertainty Modeled
WGAN-PINN (Gao et al., 2021)	PDE residual penalty	Distributional output via GAN latent z
Physics-conformal prediction (Gopakumar et al., 6 Feb 2025, Yu et al., 17 Sep 2025)	Residual nonconformity	Distribution-free coverage for residuals
Bayesian PINN with error-bounds (Flores et al., 9 May 2025, Ramirez et al., 7 Jan 2026)	Residual-based noise/likelihood	Heteroscedastic + epistemic
Physics-informed GP priors (Spitieris et al., 2022)	Priors/discrepancy	Joint parameter, state, and model-form
Interval/fuzzy PINNs (Fuhg et al., 2021)	Interval/fuzzy fields	Non-probabilistic envelopes
Physics-informed IFT (Alberts et al., 2023)	Functional prior, energy	Bayesian field/posterior (nonparametric)
MC X-TFC (Florio et al., 2024)	Resid