Physics-Informed Neural Framework

• Physics-informed neural frameworks are methods that embed physical laws and PDE constraints into neural networks to improve simulation and prediction accuracy.
• They employ diverse strategies such as PINNs, operator learning, and domain decomposition to balance data-driven and physics-based loss functions.
• These frameworks apply to dynamical systems, geophysics, power systems, and materials science, offering robust performance even with sparse or noisy data.

A physics-informed neural framework integrates domain knowledge—typically in the form of physical laws, partial differential equations (PDEs), or operator constraints—into the structure, training objectives, and inductive biases of neural network models. This approach addresses the limitations of purely data-driven methodologies for simulating, predicting, or inferring properties of complex physical systems, especially under scarcity of labeled observations or where high-fidelity simulation is computationally prohibitive. Such frameworks are broadly extensible, encompassing not only the canonical PINN approach but also operator-learning models, hybrid architectures, Bayesian and evidential inference, domain decomposition, and applications to forward, inverse, and optimal control settings.

1. Core Principles and Foundational Loss Structures

The central idea underlying a physics-informed neural framework is to embed the governing equations of a physical system into the architecture or loss function of neural networks. The prototypical approach is the Physics-Informed Neural Network (PINN), in which a neural network $u_\theta(x)$ is fit not only to available data but also to minimize the residuals of the underlying physics (e.g., PDEs, ODEs) evaluated at collocation points via automatic differentiation: $$\mathcal{L}(\theta) = \frac{1}{N_{\text{data}}} \sum_{i=1}^{N_{\text{data}}} |u_\theta(x_i) - y_i|^2 + \lambda \frac{1}{N_{\text{phys}}} \sum_{j=1}^{N_{\text{phys}}} |\mathcal{N}[u_\theta](x_j)|^2$$ where $\mathcal{N}$ is the relevant differential operator (Misyris et al., 2019, Rodriguez et al., 2024).

Advanced frameworks generalize this principle by incorporating domain decomposition (partition of unity), reduced-order and operator learning, variational or energetic formulations, or uncertainty quantification. Physics-based constraints can be embedded through pointwise residual penalties, as global conservation laws, or via direct optimization of variational objectives.

2. Architectural Variants and Operator Learning Extensions

Physics-informed neural frameworks encompass a spectrum of neural architectures. Classic PINNs utilize fully connected multilayer perceptrons, but many recent works have extended this to include convolutional architectures, recurrent models, attention-based networks, and operator-learning formulations:

• Convolutional PINNs and Operators: Physics-informed convolutional neural networks (PICN) leverage shallow convolutional/deconvolutional architectures, with fixed convolutional filters encoding finite-difference stencils for differential operators. This approach is tailored for efficiently capturing high-frequency content, handling irregular domains via interpolation, and rapid convergence, particularly when data is scarce (Shi et al., 2022).
• Sequence and Transformer Architectures: PINNsFormer replaces classical MLPs with Transformer encoders/decoders to model temporal dependencies via multi-head attention, pseudo-sequence generation, and a sequential physics-based loss function. The Wavelet activation anticipates Fourier decompositions, improving high-frequency representational efficacy (Zhao et al., 2023).
• Neural Operator Frameworks: Physics-informed operator models (PINO, PICNO, PI-DeepONet) learn mappings from input functions (e.g., initial/boundary data, coefficient fields) to solution fields, using architectures like Fourier Neural Operators, convolutional neural operators, or Deep Operator Networks. Loss functions enforce operator constraints not merely pointwise but over function spaces, often employing spectral or convolutional kernels (Rosofsky et al., 2022, Ma et al., 22 Jul 2025, Karampinis et al., 7 Nov 2025).
• Reduced-Order and Discretized Models: Discretized-physics-informed neural networks (DisPINN) first discretize the governing equations (e.g., via finite differences or Galerkin projection to a latent subspace), then form a neural surrogate for the reduced coordinates, and incorporate residuals of the discrete system into the loss (Halder et al., 2023).
• Evidential and Uncertainty-Aware Architectures: Evidential PINNs reformulate the physics-informed loss in terms of probabilistic hyperparameters, inferring both predictive means and uncertainties, with closed-form expressions for the marginal likelihood and information-theoretic regularization terms (Tan et al., 27 Jan 2025). Bayesian and variational approaches are also employed in operator learning and parameter inference (Myers et al., 2 Feb 2026).

3. Loss Function Design and Physics Integration

Integration of physical constraints is realized through customized loss functions, frequently combining data-driven and physics-based penalties:

• Pointwise Residuals: Losses penalize deviations from the governing equations at selected points, using automatic differentiation to evaluate derivatives (Misyris et al., 2019, Antonelo et al., 2021).
• Operator or Integral Residuals: For integral or fractional operator problems, losses can incorporate quadrature-based evaluations of operators (e.g., Fredholm or Volterra) via tensor–vector product techniques (Aghaei et al., 2024).
• Energetic or Variational Principles: In energetics-based PINNs (e.g., for flexoelectricity), the loss is defined as a saddle-point of the total potential energy, with min–max optimization over primal (e.g., displacement) and dual (e.g., potential) networks. Additional variational losses enforce stationarity constraints for robust parameter identification (Moon et al., 13 Jun 2025).
• Partitioned/Domain-Decomposition Losses: In POU-PINNs, spatial subdomains and their parameters are learned jointly with the global solution, with soft partition-of-unity weights and penalties to enforce unity and PDE consistency (Rodriguez et al., 2024).
• Confidence and Uncertainty Penalties: Information-theoretic regularizers (e.g., KL divergence between inferred and reference distributions) and closed-form coverage calibration enhance reliability of uncertainty estimates (Tan et al., 27 Jan 2025).

4. Applications and Demonstrated Impact

Physics-informed neural frameworks have been deployed across a wide variety of domains and problems:

• Dynamical Systems and Control: PINC and operator-learning approaches are used for surrogate modeling, long-range rollout, and real-time control in nonlinear ODE systems, including the Van der Pol oscillator, four-tank systems, and large-scale power grids. Efficiency far exceeds classical time integration, enabling embedding into high-throughput applications like model predictive control (Antonelo et al., 2021, Karampinis et al., 7 Nov 2025).
• Power Systems: PINNs and PI-DeepONets enable rapid state estimation, parameter identification, and transient prediction for synchronous machines and composite networks—achieving 28–87× speed-up and high precision (relative $L_2$ error $1.34\times10^{-2}$ for rotor angle prediction) (Misyris et al., 2019).
• Wave Propagation and Geophysics: Physics-informed operator models (PICNO) and PINNs-present substantial reductions in predictive $L_2$ error (up to 53%; e.g., from 0.50 to 0.23 for high-frequency geophysical wavefields) relative to purely data-driven neural operators (Ma et al., 22 Jul 2025, Rosofsky et al., 2022).
• Cosmology and Astrophysics: PINN-based emulators for baryonic inpainting in hydrodynamic simulations integrate analytic relations (e.g., SHMR) and KL divergence penalties to reproduce mean trends and scatter, enabling reconstruction of the full baryonic property set with improved accuracy and preserved physical structure (Dai et al., 2023, Myers et al., 2 Feb 2026).
• Materials and Multiphysics Modeling: Unified PINN-DEM frameworks handle forward and inverse solutions of high-order PDEs (e.g., in flexoelectricity) with energy-based losses and robust recovery of material parameters, validated against mixed finite element methods (Moon et al., 13 Jun 2025).
• Integral and Fractional Operator Problems: The PINNIES framework delivers fast tensorized quadrature for Fredholm/Volterra/fractional equations and optimal control, yielding MAE as low as $10^{-6}$ and outperforming automatic differentiation and competitor packages (Aghaei et al., 2024).

5. Computational Efficiency, Generalization, and Limitations

Physics-informed frameworks exhibit several computational and generalization benefits:

• Sample Efficiency: The physics constraint dramatically reduces dependence on labeled data. For example, PINNs for SMIB swing equations achieve sub-percent error with as few as 40 training points (Misyris et al., 2019); DisPINNs reach $O(10^{-2})$ error with 1–3 points, outperforming vanilla data-driven networks by up to an order of magnitude (Halder et al., 2023).
• Inference Acceleration: Evaluating a trained PINN or operator surrogate is typically several orders of magnitude faster than classical solvers (e.g., $0.004$ s vs $0.35$ s for SMIB swing equations at arbitrary time).
• Robustness to Sparse/Noisy Data: Incorporation of known physics grants improved generalization to unseen settings, high-resolution extrapolation, and calibration even when observational data are limited or noisy (Tan et al., 27 Jan 2025, Sarabian et al., 2021).
• Limitations: Frameworks can suffer from increased training cost, especially with large numbers of collocation points or stiff/nonlinear regimes. Enforcing sharp discontinuities or handling discrete events requires extension (e.g., POU-domain decomposition or hybrid retraining strategies). Some frameworks require careful hyperparameter tuning (loss weights, partition cardinality). Scalability to high-dimensional systems may necessitate operator learning or domain decomposition (Rodriguez et al., 2024, Halder et al., 2023).

6. Recent Innovations: Domain Decomposition and Uncertainty Quantification

Recent works have focused on expanding flexibility, reliability, and transparency:

• Partition of Unity and Mixtures of Experts: POU-PINN discovers spatial subdomains with distinct physics or parameters in an unsupervised fashion, leading to improved accuracy for PDEs with sharp coefficient variations (e.g., conductivity in porous media, ice subdomains in glaciology). Errors decreased from $10^{-2}$ to $10^{-6}$ compared to conventional PINNs, and convergence is accelerated by up to $5\times$ (Rodriguez et al., 2024).
• Evidential and Bayesian Physics-Informed Inference: E-PINN integrates uncertainty quantification by learning higher-order evidential priors over model outputs and PDE parameters, providing closed-form predictive variances and empirically reliable coverage probabilities (e.g., ECP=0.93–0.96 on 1D/2D inverse problems) (Tan et al., 27 Jan 2025). Bayesian PINNs and operator models (e.g., for galactic potentials) further deliver posterior credible intervals and calibration (Myers et al., 2 Feb 2026).
• Hybrid and Empirical-Physics Regularization: Self-supervised frameworks (SPINN) and empirical-physics-weighted losses adaptively balance data and physics terms, improving extrapolation and estimation even with very limited data and providing robust error bounds (Pirayeshshirazinezhad, 7 Sep 2025).

7. Software Libraries and Reproducibility

To facilitate adoption, open-source frameworks such as IDRLnet (Peng et al., 2021) and PINNIES (Aghaei et al., 2024) provide extensible software for PINN research and applications. These packages incorporate modular abstractions for geometric domains, data integration, neural architectures, physics-informed and empirical loss terms, quadrature, and optimization. Tutorials, code, and benchmark data enable straightforward extension to novel physical problems and reproducibility of published results.

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Physics-informed neural frameworks represent a rapidly growing paradigm for combining the representational power of neural networks with the interpretability and constraint of physical law. Their flexibility, efficiency, and generalization capabilities have been demonstrated across a spectrum of scientific and engineering domains, with ongoing research addressing scalability, robustness, domain decomposition, and uncertainty quantification (Misyris et al., 2019, Halder et al., 2023, Moon et al., 13 Jun 2025, Rosofsky et al., 2022, Tan et al., 27 Jan 2025, Rodriguez et al., 2024, Ma et al., 22 Jul 2025, Karampinis et al., 7 Nov 2025, Myers et al., 2 Feb 2026, Aghaei et al., 2024).