Physics-Informed Gaussian Process Method

• Physics-Informed Gaussian Process methods are defined by embedding governing physical laws directly into GP formulations to yield physically consistent and data-efficient predictions.
• They integrate analytical knowledge from differential equations with data-driven techniques to enhance inverse problem solving, system identification, and control applications.
• These methods deliver practical benefits such as closed-form uncertainty quantification, improved scalability via eigenfunction expansions, and robust fusion of heterogeneous sensor data.

A physics-Informed Gaussian Process (PI-GP) method rigorously embeds governing physical laws—typically differential equations, constitutive relations, or conservation laws—directly into the formulation of Gaussian Process priors, kernels, or learning objectives. This family of techniques augments or replaces purely data-driven nonparametric modeling with analytic knowledge from physics, yielding models that are more data-efficient, physically consistent, and capable of uncertainty quantification over both observable and latent physical states. The approach finds critical application in inverse problems, system identification, control, and forecasting across domains as diverse as structural health monitoring, acoustic sensing, power systems, and fluid mechanics.

1. Mathematical Foundations and Model Formulation

A PI-GP constructs a Gaussian Process prior that is closed under the relevant differential operators representing the physics of interest. Consider the Euler–Bernoulli beam equation as a canonical example:

$EI \,\frac{d^4 u(x)}{dx^4} = q(x)$

where $u(x)$ is the transverse deflection, $q(x)$ is the applied load, and $EI$ is bending stiffness. A PI-GP places a joint Gaussian-process prior on the deflection $u(x)$ and all observable/latent quantities derivable by linear operations: rotations ($r = u'$), strains ($\epsilon = -z u''$), moments ($m = -EI u''$), shear forces, and loads. If

$$u(x)\sim\mathcal{GP}\bigl(0,\, k_{uu}(x,x';\theta)\bigr)$$

then all other fields are also GPs, and cross-covariances are computed by applying the relevant derivatives to $k_{uu}$ (e.g., $$k_{qq}(x,x')=EI^2\,\frac{\partial^4}{\partial x^4}\frac{\partial^4}{\partial x'^4}k_{uu}(x,x')$$). This closure under linear operators generalizes to elliptic, parabolic, and wave PDEs, including Helmholtz, Poisson, or wave equations, as well as to domains beyond beams, such as vibro-acoustics (Tondo et al., 2023, Albert, 2019).

In multi-output settings,

$$\mathbf{Y}(x) = [u(x), r(x), \epsilon(x), m(x), v(x), q(x)]^T$$

the block covariance is assembled using the above analytical derivations, forming a kernel matrix $\mathbf{K}_p(x,x';\theta, EI)$ that encapsulates the entire physical model.

2. Incorporating Physics in Kernels and Priors

2.1 Operator-Induced and Eigenfunction Kernels

PI-GPs can enforce physics via:

• Kernel Differentiation: When the operator is linear, GPs are closed under differentiation and multiplication-by-constants, enabling covariance and cross-covariance computation for any linear-functional of the process (Tondo et al., 2023, Pförtner et al., 2022, Albert, 2019).
• Eigenfunction Expansions: Embedding domain and boundary conditions directly in the kernel via Laplacian eigenfunction expansions guarantees that all samples respect physical BCs, as shown in plate wave propagation (e.g., for acoustic emission mapping with built-in Neumann or Dirichlet BCs) (Jones et al., 2022).
• Green's Function or Analytical Kernels: For certain PDEs (e.g., Helmholtz or Poisson), the covariance is chosen as the appropriate Green’s function, ensuring all prior samples are solutions to the homogeneous equation (Albert, 2019).

2.2 Data-Driven and Hybrid Physics Priors

Physics may be incorporated via:

• Monte Carlo or Multilevel MC Priors: Statistical moments (mean, covariance) are estimated from ensembles of physics-based simulation runs (e.g., solving stochastic PDEs under random parameterizations), yielding empirical, typically non-stationary kernels (Yang et al., 2018).
• Surrogate Model-Informed Means: For control/optimization, the GP mean and structure may be informed by a fast surrogate or low-order physics-based model, reducing the need for extensive data (Hanuka et al., 2020, Hanuka et al., 2019, Harp et al., 2 Jan 2025).
• Differential Equation Collocation: Soft or hard constraints from the governing PDE can be encoded as pseudo-observations or as additional likelihood terms in the GP, penalizing violations of the physics (Long et al., 2022, Pförtner et al., 2022).

3. Training, Inference, and Uncertainty Quantification

3.1 Bayesian Conditioning

Conditioning the joint multi-channel PI-GP on noisy sensor data entails standard GP regression formulas augmented with channel-specific noise variances, such that

$$\log p(\mathbf{y}|\psi) = -\frac{1}{2}\mathbf{y}^T\mathbf{K}^{-1}\mathbf{y} - \frac{1}{2}\log|\mathbf{K}| - \frac{n}{2}\log 2\pi$$

where $\psi$ contains kernel hyperparameters, physical parameters (e.g., $EI$), and channel noise levels. Posterior over model parameters (e.g., stiffness, diffusivity) is sampled via MCMC or optimized via maximum marginal likelihood (Tondo et al., 2023). The posterior predictive distribution is Gaussian (or a mixture if hyperparameters are integrated out), from which means and variances at arbitrary spatial or temporal points—and thus uncertainty quantification—are computed in closed form.

3.2 Physical Constraints Satisfaction

PI-GPs can guarantee that predictions exactly satisfy physical constraints if the prior and observation model are constructed accordingly. For example, a covariance constructed from eigenfunctions that satisfy the domain BCs ensures that all prior and posterior samples inherit those boundary behaviors (Jones et al., 2022); kernel differentiation guarantees closure under linear constraints (Tondo et al., 2023, Pförtner et al., 2022). For nonlinear or incomplete physical knowledge, the constraints may be “soft,” factored into the likelihood as in AutoIP (Long et al., 2022).

4. Applications and Empirical Performance

4.1 System Identification and Structural Health Monitoring

In structural monitoring, the PI-GP regresses distributed parameters such as $EI$ and fuses heterogeneous sensor data (deflection, rotation, strain) with physically-coupled covariance structure. Anomalies or damage are detected by computing the Mahalanobis distance of the inferred stiffness distribution from the nominal reference, enabling localization and quantification of structural changes (Tondo et al., 2023). Experimental results demonstrate sub-percent accuracy in $EI$ estimation and the ability to down-weight faulty sensors automatically.

4.2 Physics-Informed Extrapolation with Sparse Data

Embedding BCs via eigenfunction or Green’s function kernels leads to substantial reductions in required data for high-fidelity interpolation—especially under sparse or partial-coverage sensor deployments—outperforming standard GPs by factors of 2× in MSE in data-scarce regimes (Jones et al., 2022). For acoustic or vibro-acoustic systems, field reconstructions respect all physical constraints by construction and allow simultaneous inverse estimation of material properties and source strengths (Albert, 2019).

4.3 Safe Control and Model Predictive Control

In dynamics and control, PI-GPs are used to encode ODE or PDE system dynamics within the GP prior, allowing closed-form trajectory inference constrained by the physics (Lepp et al., 30 Apr 2025, Tebbe et al., 20 Nov 2025). Quadratic costs and box constraints are handled via conditioning and Hamiltonian Monte Carlo post-processing for open-loop guarantee of constraint satisfaction; convergence and stability properties can be inferred analytically from the Bayesian setting.

4.4 Data-Driven Surrogates and Active Learning

When obtained from physics-based simulations, ensemble-based PI-GPs can obviate the need for hyperparameter training, yielding nonparametric surrogates for function approximation and optimal sensor placement via active learning that targets maximum predicted variance (Yang et al., 2018).

5. Algorithmic and Computational Considerations

• Scaling: Classical GP inference is limited by $O(n^3)$ scaling, but basis-expansion PI-GPs (e.g., eigenfunction expansions) reduce complexity to $O(nm^2)$ for $m$ basis functions, and are tractable for $n\sim 10^4$ with moderate $m$ (Jones et al., 2022).
• Numerical Implementations: Eigenvalue problems for complex domains are addressed via sparse-matrix solvers (e.g., ARPACK), with BCs enforced via ghost-node techniques or boundary modifications to finite-difference stencils (Jones et al., 2022).
• Robustness: By crafting covariances that respect physics, uncertainty is minimized in well-constrained regions and inflated only where neither data nor physics provides guidance, yielding well-calibrated predictive variances (Tondo et al., 2023).

6. Comparative Analysis and Impact

PI-GP frameworks unify several previously separate approaches:

• They generalize classical numerical solvers for PDEs (collocation, Galerkin/FEM, spectral methods) in the sense that the GP posterior mean coincides with the solution from these methods for a suitable choice of kernel and observation operator; the posterior variance quantifies discretization, modeling, and measurement uncertainty in a principled way (Pförtner et al., 2022).
• Active learning and optimal informativeness (sensor placement) can be achieved by maximizing entropy or minimizing conditional entropy, leveraging physics-based cross-covariances to propagate information across observable and latent physical states (Tondo et al., 2023).
• Hybridization with data-driven models allows for robust identification and control when only partial or uncertain physical constraints are available, providing resilience to model misspecification.

7. Extensions, Challenges, and Outlook

PI-GPs admit generalization beyond linear physics and stationary domains:

• Nonlinear differential operators can be handled via variational methods with “soft” physics likelihoods, as in latent-source AutoIP frameworks (Long et al., 2022).
• Inclusion of multi-output, mixed-fidelity sensors, and incorporation of unknown physical parameters as hyperparameters in the kernel, allow simultaneous field reconstruction, parameter identification, and uncertainty quantification within a consistent Bayesian framework (Tondo et al., 2023, Tondo et al., 2023).
• Bottlenecks include scaling to high dimensions, representing highly non-stationary or anisotropic physics, and extension to nonlinear, mixed, or incomplete physics, which are active areas of research.

Physics-Informed Gaussian Process methods thus provide a principled, probabilistic, and physically consistent extension of classical regression, system identification, and inverse problem frameworks, achieving data efficiency, robust uncertainty quantification, and rigorous enforcement of laws of physics in a nonparametric machine learning context (Tondo et al., 2023, Pförtner et al., 2022, Jones et al., 2022, Albert, 2019, Tondo et al., 2023).