Physics-Informed Methods in Machine Learning

Updated 27 November 2025

Physics-informed methods are techniques that integrate governing physical laws into learning models to enforce consistency and improve generalization.
They implement composite loss functions that balance data fitting with physical constraints, such as PDE residuals and conservation laws.
Applications include surrogate modeling, uncertainty quantification, and field inversion, driving robust performance in scientific computing.

A physics-informed method refers to any machine learning or statistical approach in which knowledge of the governing physical laws is directly embedded into the inference, learning, or optimization process. This can occur through the explicit inclusion of partial differential equations (PDEs), conservation laws, or constitutive relations as constraints or penalty terms in the training objective, or through the design of feature spaces and priors that reflect established physical properties. Physics-informed methods have become a dominant paradigm in scientific computing, computational physics, and data assimilation, yielding solution techniques that can efficiently leverage both data and physics for better interpretability, robustness, efficiency, and generalization.

1. Mathematical Frameworks for Physics-Informed Methods

Physics-informed approaches span a wide range of mathematical formulations and algorithmic frameworks, including neural-network-based surrogates, kernel methods, field-theoretic models, and reduced-order stochastic expansions. The common underlying structure is a composite loss or objective functional that enforces consistency with physical laws while assimilating data.

Prototypical Objective Structure

$\min_{w} \; L_{\rm data}(w) + \lambda L_{\rm phys}(w) + \alpha L_{\rm reg}(w)$

$L_{\rm data}(w)$ : Fit to observed or measured data.
$L_{\rm phys}(w)$ : Quantified violation of the governing physical law (e.g., PDE residual, conservation constraint).
$\lambda$ : Weighting of the physics loss.
$L_{\rm reg}(w)$ : Regularization promoting smoothness or other a priori properties.

In neural network surrogates, this is typically realized as PINNs or their extensions (Raissi et al., 29 Aug 2024), while in Gaussian processes or kernel methods, it appears as penalty or constraint terms directly within the RKHS norm (Doumèche et al., 20 Sep 2024). Variational or field-theoretic approaches formulate the prior over admissible solutions as a probability distribution sharply peaked on the space of physically admissible fields (Alberts et al., 2023).

2. Physics-Informed Neural Networks (PINNs) and Extensions

PINNs constitute a prominent class of methods in which fully connected deep networks approximate solution fields or parameters, with physics enforced by penalizing PDE residuals via automatic differentiation. The architecture, loss structure, and optimization workflow are standardized across many scientific domains.

Canonical PINN Loss

$L(\theta) = \frac{1}{N_{\rm data}}\sum_{i=1}^{N_{\rm data}} |u(x_i, t_i; \theta) - u_i|^2 + \frac{1}{N_{\rm coll}}\sum_{j=1}^{N_{\rm coll}} |\partial_t u(x_j^c, t_j^c; \theta) + \mathcal{N}[u(x_j^c, t_j^c; \theta)]|^2$

where $\mathcal{N}$ is the nonlinear PDE operator (Raissi et al., 29 Aug 2024).

Training and Extensions

Training employs staged optimizers (e.g., Adam for exploration, L-BFGS for fine-tuning).
Extensions include adaptive loss weighting, domain decomposition (CPINN, XPINN), variational hp-refinement, causality-preserving losses, fractional/stochastic operators, and theoretically grounded convergence analysis (Raissi et al., 29 Aug 2024, Bi et al., 2 Aug 2025).
For dynamic domains with moving interfaces, extended PINN architectures integrate level-set representations and physically tailored loss terms, achieving order-of-magnitude improvements in accuracy and convergence rates (Bi et al., 2 Aug 2025).

High-Dimensional and Multiscale Cases

PINNs have been effectively applied to high-dimensional integrable systems—Kadomtsev–Petviashvili equations and reduced KP—using up to 11-layer networks with architecture and loss functions scaling robustly with problem dimension (Miao et al., 2021).

3. Field Inversion, Surrogate Modeling, and Reduced-Order Approaches

Physics-informed field inversion addresses problems of correcting low-fidelity models or extracting spatially distributed correction fields under severe data sparsity.

Physics-Informed Field Inversion (PIFI)

Total loss: $L_{\rm total}(\beta) = L_{\rm data}(\beta) + \lambda L_{\rm phys}(\beta)$
Dense “virtual observations” from the physics loss regularize sparse or truncated data, efficiently constraining posterior uncertainties (Ugur et al., 23 Sep 2025).
Posterior correction fields recover spatial truth in a single realization, with only modest computational overhead (2–6%).

Surrogate Parameterization: The PICKLE Method

Utilizes conditional Karhunen–Loève (KL) expansions for parameter fields and states, with the basis and covariance matrices determined by physics and measurement locations, not mesh size.
The physical constraint is enforced in a reduced parameter space: optimization is conducted over expansion weights for the truncated KL bases (Yeung et al., 2021).
Shows computational scaling nearly linear in mesh size ( $N_{FV}^{1.15}$ ), vastly outperforming MAP Bayesian inversion ( $N_{FV}^{3.28}$ ), particularly in large-scale subsurface flow with variable Dirichlet and Neumann boundary conditions.
Once trained for one set of Dirichlet boundary conditions, the method enables rapid adaptation to new Dirichlet data by updating only the surrogate means, with no need for retraining the covariance or KL bases.

4. Statistical Learning Theory and Kernel Perspectives

Physics-informed methods in kernel regression replace generic regularization with explicit PDE-based priors.

Physics-Informed Kernel Learning (PIKL)

Minimizes a risk functional: $R_n(f) = \sum\nolimits_{i=1}^n |f(X_i)-Y_i|^2 + \lambda_n \|f\|_{H^s}^2 + \mu_n \| \mathcal{D}f \|_{L^2}^2$
The RKHS is induced by the PDE regularizer, and Fourier-based spectral methods provide tractable closed-form predictors (Doumèche et al., 20 Sep 2024).
Theoretical results show error rates that can be parametric ( $O(n^{-1} \log^3 n)$ ) when the true function exactly satisfies the physics, compared to the slower generic Sobolev rates for pure data-driven learning.
PIKL demonstrates superior accuracy and robustness to noise in both hybrid modeling and classical PDE solving compared to PINNs and finite-difference schemes.

Complexity-Dependent Error Analysis

Soft-penalized physics-informed estimators and hard-constrained empirical risk minimizers have identical rates up to constants under minimal assumptions.
Incorporation of physics reduces the effective dimension of the hypothesis class, resulting in statistically significant gains (e.g., rate improvement from $O(\sqrt{p/n})$ to $O(\sqrt{p'/n})$ where $p'$ is the dimension of admissible solutions) (Marcondes, 27 Oct 2025).
The framework is applicable in any convex function class constrained by linear physical operators.

5. Physics-Informed Uncertainty Quantification and Bayesian Field Theory

In Bayesian formulations, physics is encoded as the prior—or as an energy functional—in the space of fields.

Physics-Informed Information Field Theory (PIFT)

The prior over fields is given by $p(\phi) \propto \exp[-\beta U[\phi]]$ , where $U[\phi]$ enforces the PDE weakly, and $\beta$ quantifies model confidence (Alberts et al., 2023).
The posterior quantifies both aleatoric (data/measurement) and epistemic (model-form) uncertainty. Hyperparameters (e.g., physical coefficients, $\beta$ ) are learned from data.
The approach remains mesh-independent and can accommodate multi-modal posteriors when the physical problem is ill-posed or nonunique.
Stochastic gradient Langevin dynamics (SGLD) is used for sampling posterior distributions over fields and parameters.

Gaussian-Process and Multifidelity Approaches

Physics-informed Kriging/CoKriging (PhIK/CoPhIK) uses ensemble means/covariances from simulator outputs, blending low- and high-fidelity data with theoretical error bounds on the degree to which physical constraints are satisfied (Yang et al., 2018).
Bifidelity extensions allow rapid accumulation of GPR priors via anchored low- and high-fidelity simulations, achieving orders-of-magnitude computational savings while preserving physical structure (Yang et al., 2018).

6. Specialized Variants and Challenges

Physics-informed methodologies have been extended for diverse goals:

Field inversion with mesh-free/mesh-based surrogates: Surrogates parameterized via global bases (KL expansions, wavelets (Han et al., 11 Aug 2025)) or mesh-based models, with the mesh used only for residual evaluation.
Feature construction for supervised learning: Nonlinear, dimensionally-consistent, physics-informed feature maps allow interpretable regression/classification and discovery of governing equations (Lampani et al., 23 Apr 2025).
Control and data assimilation: Coupling physics-informed learning with control (model predictive control) and large-scale data assimilation in high-noise environments (Ma et al., 2023).
Physical solution reconstruction: Explicit constraint force methods create interpretable and robust solution reconstructions even when model physics are misspecified, ensuring physical quantifiability of the correction and insensitivity to loss-design choices (Rowan et al., 8 May 2025).

7. Outlook and Current Limitations

Physics-informed methods offer theoretically grounded, practically robust routes to assimilating data and physics for a wide spectrum of scientific and engineering problems. Key advantages include mesh-independence, parameter-space dimensionality reduction, robust uncertainty quantification, and adaptability to new boundary or input conditions.

Challenges remain:

Scalability to very high dimensions (both in parameters and space-time).
Handling rough, non-Gaussian, or discontinuous fields, especially in surrogates relying on smooth spectral basis expansions.
Dependence of accuracy and computational efficiency on user-selected kernels, regularization weights, or basis truncation choices.
The need for efficient training or inference strategies for extremely large datasets, parameter spaces, or for forward/inverse tasks with severe ill-posedness.

Ongoing research continues to extend the theoretical foundations, algorithmic efficiency, and practical applicability of physics-informed methodologies—spanning deep-learning, kernel methods, surrogate modeling, Bayesian inference, and hybrid data–physics integration (Raissi et al., 29 Aug 2024, Doumèche et al., 20 Sep 2024, Marcondes, 27 Oct 2025, Yeung et al., 2021, Ugur et al., 23 Sep 2025, Alberts et al., 2023).