LC-prior GP: Law-Constrained Gaussian Process

• LC-prior GP is a probabilistic framework that integrates physical laws into its prior via tailored kernel and operator methods.
• It employs strategies like operator-based constructions, Green’s function kernel tailoring, and latent variable integrations to enforce constraints from PDEs and conservation principles.
• This approach enhances surrogate modeling, uncertainty quantification, and generalization in complex scientific and engineering applications.

A Physical Law-Corrected Prior Gaussian Process (LC-prior GP) is a probabilistic modeling framework in which a Gaussian process (GP) prior is not selected arbitrarily but is constructed or modified to strictly incorporate known physical laws, such as conservation principles, partial differential equations (PDEs), symmetry, or invariance properties. This approach facilitates physically consistent predictions, improved generalization, and interpretable uncertainty quantification in settings ranging from computational physics to scientific machine learning and surrogate modeling for complex systems.

1. LC-prior GP: Principles and Rationale

The distinguishing principle of an LC-prior GP is the explicit modification or construction of the GP prior to enforce physical constraints on the space of admissible functions. This is typically achieved by:

• Restricting the GP’s sample paths to exactly or approximately solve a system of linear (or nonlinear) PDEs, ODEs, or operator equations governing the domain of interest.
• Modifying the kernel (covariance function) to reflect key physical properties (e.g., regularity, periodicity, conservation, or spectral structure).
• Embedding the prior in a function space consistent with physical mechanisms, such as using the Green’s function or impulse response of a linear system to “filter” the base kernel.

The rationale is that purely data-driven priors, while flexible, may yield predictions incompatible with scientific laws, poor extrapolation, and loss of physical interpretability. LC-prior GP methodologies aim to guarantee that GP realizations are physically admissible and that hyperparameters possess immediate physical meaning.

2. Algorithmic and Analytical Construction of Law-Corrected Priors

Several algorithmic strategies for constructing LC-prior GPs have been developed:

a) Operator-Based Construction

As described in "Algorithmic Linearly Constrained Gaussian Processes" (Lange-Hegermann, 2018), one encodes the physical law as a linear operator matrix $A$ and parametrizes the solution space $sol_A = \{f:\, Af=0\}$ by finding a right kernel operator $B$ so that $f = Bg$ for some GP $g$. The GP prior is then "pushed forward": $BG(g) = GP( B\mu(x), Bk(x,x')B^T )$, where $\mu(\cdot)$ and $k(\cdot,\cdot)$ are mean and covariance functions of the latent GP. Grӧbner basis methods enable the symbolic parameterization of $sol_A$ for differential and operator equations, ensuring computational efficiency and exact satisfaction of physical constraints.

b) Kernel Tailoring via Green’s Functions and Spectral Methods

Building on physical invariance of Gaussian random fields under linear operators, kernels are tailored to satisfy the homogeneous version of the governing equation $L[u]=0$ (Albert, 2019). Fundamental solutions or Green’s functions are used for convolution kernels, or eigenfunction expansions up to prescribed boundary conditions. For example, the Laplace equation in 2D may yield a kernel via a polar harmonic expansion (e.g., $$k(x,x';s)=\sum_m (rr'/s^2)^m \sigma_m^2\cos(m(\theta-\theta'))$$), where $s$ is a characteristic length.

The EPGP framework (Härkönen et al., 2022) constructs kernels using the Ehrenpreis–Palamodov principle, representing solutions as integrals over the characteristic variety of the PDE and leveraging symbolic computation to ensure that every GP realization solves the system exactly.

c) Coupling with Data-Driven Models and Surrogates

The LC-prior GP paradigm is naturally adapted to reduced-order surrogate modeling for parametric PDEs. In "Gaussian process surrogate with physical law-corrected prior for multi-coupled PDEs defined on irregular geometry" (Tang et al., 1 Sep 2025), high-dimensional PDE solutions are first projected onto a low-dimensional proper orthogonal decomposition (POD) basis. GP surrogates are then constructed for the POD coefficients, and physical laws are incorporated by learning correction functions that adjust the GP mean to minimize physics-based residuals.

d) Latent Variable and Function-Space Integrations

Advanced architectures, such as LVM-GP (Feng et al., 30 Jul 2025), interpolate between deterministic feature mappings and GP priors in a high-dimensional latent space modulated by confidence-aware weighting, with the decoder forming a stochastic solution map constrained by PDE residual minimization.

Function-space prior regularization (see FSP-Laplace (Cinquin et al., 18 Jul 2024)) selects a GP prior over output functions (rather than weights) of neural networks. The prior, designed to encode physical structure, imposes an RKHS norm penalty, and learning is recast as finding a 'weak mode' in function space.

3. Physical Law Correction Mechanisms

Physical law correction can be realized in several ways:

• Hard Constraints: Constructing priors such that all realizations exactly satisfy the governing law, e.g., kernels restricted to the solution space of a PDE (EPGP).
• Soft Constraints: Incorporating loss terms that penalize deviations from the law, e.g., physics-informed neural networks or residual regularization in surrogate modeling (Tang et al., 1 Sep 2025, Feng et al., 30 Jul 2025).
• Kernel Filtering: Embedding information about the physical system via convolution of the base kernel with the Green’s function of the operator (Beckers et al., 2023).
• Latent Force Models: Augmenting state-space representations to include GP-driven latent inputs within the dynamics, followed by Kalman filtering and optimal control designs that exploit the learned force structure (Särkkä et al., 2017).

These corrections ensure that the prior expectation, covariance, or the full function space remains tightly coupled to well-posed physical models and observed constraints.

4. Hyperparameter Selection and Interpretability

A major benefit of LC-prior GP constructions is the physical interpretability of kernel hyperparameters and posterior predictions:

• Parameters such as length scales, wavenumbers, diffusivity, or reaction rates in the kernel map directly to physical quantities (Albert, 2019).
• In law-corrected priors, hyperparameters can be optimized via marginal likelihood or Bayesian estimation, obtaining physically meaningful values (e.g., optimum $k_0$ matches the physical acoustic wavenumber in Helmholtz equation regression).
• Spectral properties (e.g., eigenvalue decay exponent $\alpha$ and target expansion decay $\beta$ (Jin et al., 2021)) dictate learning curve asymptotics, sample efficiency, and represent the smoothness/complexity imposed by physical laws.

Recent work on Matern kernel optimization using physical insights and catastrophe theory (Niroomand et al., 2023) further illuminates the role of hyperparameter landscapes, revealing that oft-used half-integer values of smoothness are typically suboptimal; careful tuning may yield significant gains in mean squared error and convergence.

5. Numerical Methods, Computational Strategies, and Surrogate Modeling

The practical deployment of LC-prior GP models hinges on efficient algorithms for:

• Sparse and Algorithmic Kernel Construction: Sparse spectrum approximations (S-EPGP (Härkönen et al., 2022)), symbolic computation of differentiation matrices (RBF-FD (Tang et al., 1 Sep 2025)), and efficient matrix-free Laplace approximation in large-scale functional regularization (Cinquin et al., 18 Jul 2024).
• Dimensionality Reduction: Multi-output PDE systems are commonly dealt with via proper orthogonal decomposition to reduce the number of GP models and accelerate cross-parameter evaluation (Tang et al., 1 Sep 2025).
• Adaptive Sampling and Experimental Design: Active learning turbocharged by physically-corrected GP surrogates uses combined D-optimality and space-filling criteria to maximize equation discovery and data economy (Chen et al., 2019).
• Uncertainty Quantification: Probabilistic decoders (neural operators, DeepONet) and Bayesian inference in reduced latent spaces (HMC, variational inference) enable reliable uncertainty bands and robust posterior estimation in inverse problem scenarios (Feng et al., 30 Jul 2025, Meng et al., 2021).

6. Applications and Performance

LC-prior GP frameworks have demonstrated significant effectiveness in diverse domains:

• Physical System Inference: Recovery of latent signals from blurred (mixture) measurements in sensing (Tobar et al., 2017), time-series inference, edge detection, and heart-rate dynamics modeling.
• High-Fidelity Physics Surrogates: Surrogate modeling for reaction–diffusion, miscible flow, and incompressible Navier–Stokes equations defined on irregular domains (Tang et al., 1 Sep 2025), with systematic error reduction and improved extrapolation.
• Data-Efficient Learning of Physical Laws: Simultaneous estimation of field values, source strengths, and physical parameters in Laplace, Helmholtz, and heat equations, with hyperparameters providing direct physical insights (Albert, 2019).
• Meta-learning and Bayesian Deep Learning: Efficient uncertainty-aware modeling for inverse PDE problems, coupled multi-physics, and scientific time-series, where prior knowledge is abundant and physically motivated kernels outperform generic choices (Meng et al., 2021, Cinquin et al., 18 Jul 2024).

Performance metrics consistently indicate reduced estimation error, enhanced uncertainty quantification, and improved recovery of physical spectral content compared to standard GP and neural baseline methods.

7. Implications, Limitations, and Future Directions

The LC-prior GP paradigm advances data-driven modeling by rigorously integrating first-principles physics, enabling:

• Reliable and interpretable surrogate modeling for high-dimensional and computationally intensive parametric PDEs.
• Improved generalization and sample efficiency in settings with limited or noisy data.
• Robust solution of inverse problems with tight uncertainty bounds and physically consistent predictions.

Limitations remain: exact prior construction is most tractable for linear systems with analytically or symbolically characterizable solution spaces; general nonlinear laws, complex boundary conditions, or systems lacking full parametrization may challenge current frameworks. Efficient kernel learning on very large datasets or with complex geometry remains an active research area.

A plausible implication is that ongoing developments in symbolic computation (Grӧbner bases, Macaulay2), operator learning, and probabilistic neural architectures will further broaden the applicability and scalability of LC-prior GP models. Adaptive selection of correction points, more expressive kernel design, and integration with active learning stand out as promising directions.

This synthesis establishes LC-prior Gaussian processes as a foundational methodology in scientific machine learning for physically constrained surrogate modeling, equation discovery, and uncertainty-aware inference in complex systems.