Physics-Informed Machine Learning

• Physics-informed machine learning is an interdisciplinary approach that embeds physical laws, such as differential equations and conservation principles, into machine learning models to ensure physically plausible predictions.
• It employs methodologies like loss function augmentation, architectural modifications, and hybrid grey-box models to combine empirical data with theoretical principles.
• PIML enhances data efficiency, accelerates convergence, and offers robust uncertainty quantification for diverse applications including fluid dynamics, structural monitoring, and energy forecasting.

Physics-informed machine learning (PIML) is an interdisciplinary framework that systematically integrates physical laws or domain-specific constraints—often formalized as differential equations or conservation principles—directly into machine learning models. By embedding physics into the learning process, PIML aims to improve data efficiency, generalizability, and physical plausibility of predictions compared to conventional data-driven approaches. This paradigm unifies empirical data with first-principles knowledge, resulting in models that are both statistically robust and consistent with underlying scientific mechanisms.

1. Fundamental Principles and Theoretical Foundation

Physics-informed machine learning leverages mathematical representations of physical phenomena, predominantly via ordinary and partial differential equations (ODEs and PDEs), symmetry constraints, and conservation laws, as structural priors in the learning process. In canonical supervised learning, the regression function $f^*$ is estimated by minimizing empirical risk, often regularized by smoothness or complexity constraints:

$$R_n(f) = \frac{1}{n}\sum_{i=1}^n |f(X_i) - Y_i|^2 + \lambda_n \|f\|_{H^s(\Omega)}$$

PIML augments this with a physics-based penalty, typically expressed as a PDE fidelity term:

$$R_n(f) = \frac{1}{n}\sum_{i=1}^n |f(X_i) - Y_i|^2 + \lambda_n \|f\|_{H^s(\Omega)} + \mu_n \|\mathcal{D}(f)\|_{L^2(\Omega)}$$

where $\mathcal{D}$ denotes a differential operator encoding the physical law (e.g., Laplace, Helmholtz, Navier-Stokes), and $\mu_n$ controls the influence of the physics prior (Doumèche et al., 12 Feb 2024, Doumèche, 11 Jul 2025).

The theoretical analysis of PIML demonstrates that, for linear differential operators, this framework can be recast as a kernel learning problem, where the associated reproducing kernel Hilbert space (RKHS) is induced by the physical and smoothness constraints. Closed-form solutions, quantitative convergence rates, and error bounds can then be derived, establishing that the incorporation of physical priors accelerates convergence and improves statistical performance when the physics is approximately satisfied (Doumèche et al., 12 Feb 2024, Doumèche et al., 20 Sep 2024, Doumèche, 11 Jul 2025).

2. Methodological Strategies for Physics Integration

PIML incorporates physical principles within machine learning at multiple levels:

• Loss Function Augmentation: Physics is directly encoded into the objective via penalties on PDE, ODE, or conservation law residuals. For instance, PINNs minimize a composite loss:

  $$\mathcal{L} = \mathrm{MSE}_\mathrm{data} + \mathrm{MSE}_\mathrm{physics}$$

  where $\mathrm{MSE}_\mathrm{physics}$ measures the violation of the physical equations at sampled collocation points (Hao et al., 2022, Kapoor et al., 2023, Nasiri et al., 12 Aug 2024).

• Architecture and Representation: Physics can be “hard-wired” into neural network architectures via input/output transformations (e.g., Softmax to enforce mass fraction sum-to-one in combustion chemistry), coordinate-to-variable mappings (continuous PINNs), or field-to-field mappings (discrete PINNs for super-resolution) (Wu et al., 3 Sep 2025). Physics-informed graph neural networks (PIGNs) use graph exterior calculus to construct discrete operators related to divergence, gradient, and curl, enabling the modeling of unstructured, multiscale systems (Shukla et al., 2022).
• Hybrid and Grey-Box Models: These combine first-principles simulations with data-driven components, as in Bayesian grey-box Gaussian processes for structural health monitoring where the mean and kernel functions are derived from physics (Cross et al., 2022). Digital twins synergize physics-based models and sparse data to enable high-fidelity simulation and model calibration (Nghiem et al., 2023).
• Data Enhancement and Regularization: Generation of synthetic datasets by numerically simulating the physics (e.g., finite element methods for biomechanics) augments scarce experimental data and provides physically plausible samples (Raymond et al., 2021). Regularization strategies may also enforce boundary and initial conditions exactly or via soft penalization.

3. Model Classes and Algorithmic Instantiations

The core algorithmic frameworks in PIML include:

• Physics-Informed Neural Networks (PINNs): PINNs utilize neural function approximators whose training objective is augmented by the residual of governing equations (e.g., PDEs, ODEs). Differentiation is typically performed by automatic differentiation tools, enabling efficient calculation of residuals and derivatives even for high-order or coupled equations (Toscano et al., 17 Oct 2024, Chen et al., 2023).
• Physics-Informed Kernel Learning (PIKL): For linear physics, the risk minimization becomes a kernel ridge regression in a “physics-informed” RKHS. Fourier methods are used to efficiently compute the kernel induced by the physical prior, resulting in explicit estimators that achieve superior convergence rates, especially when the data conform closely to the governing physics (Doumèche et al., 12 Feb 2024, Doumèche et al., 20 Sep 2024, Doumèche, 11 Jul 2025).
• Graph-Based and Operator Learning: Physics-informed graph networks extend the PINN paradigm to graph-structured or multicomponent systems, supporting simulation on irregular meshes. Neural operator architectures such as DeepONet or Fourier Neural Operator (FNO) generalize the mapping from function spaces to solution spaces, incorporating physics-informed loss terms to enforce differential or integral constraints (Hao et al., 2022, Chen et al., 2023).
• Extreme Learning Machines (ELMs): Physics-informed ELMs use single hidden-layer networks with randomly assigned hidden weights, efficiently trained via least squares to satisfy ODE constraints and fit measured data—a method suitable for real-time geotechnical monitoring (Guo et al., 1 Oct 2025).

4. Applications Across Scientific and Engineering Domains

PIML has demonstrated wide applicability:

• Fluid and Solid Mechanics: Surrogate modeling of turbulent and laminar flows, super-resolution of flow fields, and parameter inference for nonlinear structural systems, including moving load simulations and structural health monitoring (Cross et al., 2022, Kapoor et al., 2023).
• Combustion: Enforcing mass/momentum/energy conservation and kinetics as soft or hard constraints in PDE-augmented networks for chemical kinetics, laminar and turbulent combustion, and inverse design of combustion systems (Wu et al., 3 Sep 2025).
• Thermodynamics and Liquid State Theory: Solving the Ornstein-Zernike equation via PINNs and neural operator networks, enabling rapid evaluation of pair correlation functions and structure factors over a range of thermodynamic states (Chen et al., 2023).
• Reservoir Engineering: Differentiable programming unifies CNN surrogates and finite-volume PDE solvers for rapid optimization of extraction rates while preventing over-pressurization in heterogeneous underground reservoirs (Pachalieva et al., 2022).
• Aerospace Engineering: Real-time spacecraft thermal simulation leverages hybrid PIML frameworks where adaptive coarse/fine mesh specification is learned via neural surrogates and physics constraints are enforced through embedded finite-difference solvers (Oddiraju et al., 8 Jul 2024).
• Energy and Mobility Forecasting: PIML models incorporating differential or linear constraints improve the reliability and adaptability of load forecasting, especially under atypical operating conditions such as mobility-driven demand shifts (Doumèche, 11 Jul 2025).
• Time Series and Control: State-space, Lagrangian, and Hamiltonian-structured neural models, augmented with Lyapunov and barrier function constraints, provide sample-efficient learning and verifiable guarantees for dynamical systems and reinforcement learning in safety-critical settings (Nghiem et al., 2023).

5. Performance, Theoretical Guarantees, and Empirical Outcomes

• Convergence and Generalizability: Theoretical analyses show that if the target function exactly or nearly satisfies the governing physics, PIML estimators can achieve parametric convergence rates ($n^{-1}$) as opposed to the slower nonparametric Sobolev minimax rates, with the effective statistical dimension driven by the kernel’s spectrum modulated by the physical prior (Doumèche et al., 12 Feb 2024, Doumèche et al., 20 Sep 2024, Doumèche, 11 Jul 2025).
• Computational Efficiency: Closed-form or explicit solutions in kernel-based PIML (e.g., PIKL) and ELM-based approaches enable substantial reductions in training time compared to deep neural surrogates. For example, PIKL has been shown to provide faster and more robust solutions than PINNs in several PDE testbeds, especially when boundary data is noisy (Doumèche et al., 20 Sep 2024).
• Empirical Results: In domains such as head-impact detection, PIML detectors achieved high F1 scores (e.g., 0.95), with significant reductions in false positives and manual workload (Raymond et al., 2021). In industrial mineral processing, PINNs incorporating ODE-based physical constraints outperformed purely data-driven models in terms of mean squared and mean relative error under data-sparse and dynamic conditions (Nasiri et al., 12 Aug 2024). In geotechnical monitoring, physics-informed ELMs yielded accurate real-time predictions that outperformed traditional finite difference schemes, especially when integrating a small number of strategically located sensor measurements (Guo et al., 1 Oct 2025).
• Uncertainty Quantification: Bayesian grey-box and kernel-based approaches enable rigorous propagation and quantification of uncertainty, essential for applications in risk-sensitive fields such as structural health monitoring and energy forecasting (Cross et al., 2022, Doumèche, 11 Jul 2025).

6. Challenges, Limitations, and Future Directions

Key ongoing challenges in PIML include:

• Physics Knowledge Selection: The identification of appropriate physical priors (e.g., constraints, symmetries, PDE forms) remains largely problem-specific and manual. Automated discovery and modularization of physical knowledge are active research directions (Meng et al., 2022).
• Optimization and Training Dynamics: PINNs often suffer from stiffness, spectral bias, and suboptimal loss landscapes arising from the joint enforcement of multiple high-order constraints. Innovations such as adaptive loss balancing, feature expansion (e.g., Fourier embeddings), and new architectures like Physics-Informed Kolmogorov-Arnold Networks (PIKANs) are being developed to mitigate these issues (Toscano et al., 17 Oct 2024).
• Benchmarking and Reproducibility: Standardized, publicly available benchmarks are lacking across many domains, impeding fair comparison of methods and reproducibility (Wu et al., 3 Sep 2025).
• Scalability and Computational Cost: Despite the promise of kernel methods and hybrid differentiation strategies, solving high-dimensional and multiscale PDEs with PIML frameworks remains computationally challenging for very large systems (Shukla et al., 2022).
• Theoretical Frontiers: There is a need for deeper theoretical analysis of overfitting, generalization, and convergence properties of nonlinear PIML methods, particularly for nonconvex loss landscapes and for models that incorporate non-differentiable constraints (Doumèche, 11 Jul 2025).

7. Impact and Outlook

Physics-informed machine learning provides a systematic pathway to integrate data-driven learning with well-established scientific principles. This results in models that are interpretable, physically plausible, robust to extrapolation, and data-efficient. The fusion of field equations, symmetry properties, and engineering heuristics with modern statistical learning has accelerated discovery, inverse design, and real-time control in domains including fluid dynamics, aerospace, materials science, geophysics, structural health monitoring, combustion, and energy systems.

The evolution of PIML continues with the development of more expressive architectures (e.g., PIKANs), operator learning methods, scalable kernel approximations, and modular frameworks for hard and soft constraint enforcement. The field increasingly relies on interdisciplinary collaboration between physical scientists, applied mathematicians, and machine learning researchers, with mutual advances driving the development of more reliable models and practical solutions for complex scientific, engineering, and industrial challenges.