Physics-Guided CNNs: Integrating Physical Laws

• Physics-Guided CNNs (PGCNNs) are neural networks that embed physical principles into architecture and loss functions for improved prediction fidelity.
• They integrate methods such as constraint-based loss, physics-aware feature guidance, and architectural rule injection to enforce known physical laws.
• PGCNNs are applied in seismic prediction, remote sensing, and inverse problems to reduce data requirements while enhancing model robustness and interpretability.

Physics-Guided Convolutional Neural Networks (PGCNNs) represent a class of neural network models that explicitly integrate physical principles, domain knowledge, or scientific rules into the architecture, loss function, or training pipeline of convolutional neural networks (CNNs). This field merges advances in data-driven machine learning with mathematical structure from the governing physical laws of the application domain, in order to improve generalization, interpretability, resilience to limited and noisy datasets, and physical consistency of predictions.

1. Foundations and Definitions

Physics-guided convolutional neural networks are neural architectures that enforce or encode known physical constraints—derived from theoretical models, experimental laws, or expert systems—within a standard CNN framework. PGCNNs can integrate physical knowledge at several levels:

• Data curation: By using only physics-consistent datasets during training (Faroughi et al., 2022).
• Loss function design: By penalizing residuals corresponding to unsatisfied physical laws, e.g., PDE residuals, dynamic system constraints (Zhang et al., 2019, Shi et al., 2022, Zhou et al., 8 Jul 2024).
• Architectural modifications: By embedding explicit physical relationships in network layers, such as finite difference filters for PDEs (Ayensa-Jiménez et al., 2020, Shi et al., 2022) or custom modules for domain rules (Gupta et al., 3 Sep 2024).
• Feature guidance and injection: By engineering intermediate representations to align with physical quantities or expert-inferred topics (Huang et al., 2021).

This approach is distinguished from purely statistical deep learning paradigms by its fundamental reliance on prior domain knowledge as a regularization agent against overfitting and as a mechanism for extending the model’s validity beyond the training distribution.

2. Core Methodologies

PGCNN methodologies can be classified by the mechanisms through which physics is incorporated:

Approach	Integration Modality	Examples/Papers
Constraint-based loss	Adds physics loss terms to train	(Zhang et al., 2019, Zhou et al., 8 Jul 2024)
Architectural rule injection	Physics layers or custom modules	(Ayensa-Jiménez et al., 2020, Gupta et al., 3 Sep 2024)
Physics-aware feature guidance	Physics-specific latent spaces	(Huang et al., 2021, Crocker et al., 2022)
Automated architecture/loss search	AutoML for physical constraints	(Zhou et al., 8 Jul 2024)

Constraint-Based Loss

A representative formulation adds a physics-informed penalty to the standard data loss: $J(\theta) = J_D(\theta) + J_P(\theta)$ where $J_D$ is the data loss (e.g., MSE between prediction and measurement) and $J_P$ penalizes deviations from governing laws, such as conservation, balance, or equation of motion terms (Zhang et al., 2019). Alternative designs use PDE-residual loss (Shi et al., 2022) or adaptive, operator-infused losses (Zhou et al., 8 Jul 2024).

Architectural Rule Injection

Physical rules can be encoded as explicit layers. For example, (Ayensa-Jiménez et al., 2020) translates differential operators into convolution filters: $$\text{1D diff operator}: \quad D_1 = \begin{bmatrix} 0 & 0 & 0 \ 0 & -\frac{1}{l} & \frac{1}{l} \ 0 & 0 & 0 \end{bmatrix}$$ In (Gupta et al., 3 Sep 2024), custom layers enforce shape, context, and redundancy constraints at the feature map or output level, using LLM-generated dynamic rule bases and explicit spatial reasoning.

Physics-Aware Feature Guidance and Injection

Intermediate CNN feature maps are directed to capture physics-derived semantic structure (e.g., topic mixtures from physical decompositions in SAR scattering (Huang et al., 2021)), which are then fused with standard features for classification or regression tasks. This can be enforced with an auxiliary loss: $$L_{sr}(y_{phy}, \phi_I) = -\sum_k y_{phy}^{k} \log \frac{\exp(\phi_I^{k})}{\sum_i \exp(\phi_I^{i})} \cdot \delta_k$$

Automated Model and Loss Search

The Auto-PICNN approach (Zhou et al., 8 Jul 2024) applies reinforcement learning and Bayesian optimization to search an operator-rich loss function and a CNN architectural space that incorporates physical differentiation kernels, boundary constraints, and weighting strategies relevant to the PDE or operator being learned.

3. Application Domains and Case Studies

PGCNNs have been applied in a variety of scientific and engineering domains, including but not limited to:

• Seismic response prediction: PhyCNNs embed dynamic equations as loss penalties for structural vibrations, providing accurate surrogates for fragility analysis in the presence of limited data (Zhang et al., 2019).
• Remote sensing and SAR imagery: Physics-guided features derived from electromagnetic scattering models are injected into mid- and high-level feature maps for improved classification robustness and interpretability under sparse annotation (Huang et al., 2021).
• Scientific inverse problems: Subsurface imaging and nucleon-nucleon potential inversion leverage physics-guided architectures and regularization, enhancing reliability even with few samples (Crocker et al., 2022, Awasthi et al., 1 Sep 2025).
• PDE and operator learning: The solution of nonlinear ODE/PDEs, including in irregular spatial domains, benefits from CNN architectures with explicitly defined finite-difference filters, boundary conditioning, and adaptive loss weighting (Shi et al., 2022, Zhou et al., 8 Jul 2024).
• Object detection in physically constrained settings: LLM-generated spatial, shape, and context rules are implemented as programmable layers for aerial vehicle detection with demonstrable false positive reduction (Gupta et al., 3 Sep 2024).

4. Performance Characteristics and Limitations

Comprehensive empirical evaluations report the following attributes of PGCNNs:

• Data Efficiency: Embedded physics constraints significantly reduce the required sample complexity for high-precision predictions in domains such as seismic risk and remote sensing (Zhang et al., 2019, Huang et al., 2021).
• Generalization and Interpretability: By encoding prior knowledge as constraints or guided features, models exhibit improved extrapolative robustness and their decisions can be more closely traced to physically meaningful cues (Huang et al., 2021, Crocker et al., 2022).
• Reduction of Overfitting: Physics-based regularization curbs the tendency of deep models to memorize noise or spurious correlations, especially when data is scarce or noisy (Faroughi et al., 2022).
• Adaptive Automation: AutoML-driven PGCNNs optimize both loss functions and architectures given specific physics contexts, regularly surpassing hand-tuned baselines (Zhou et al., 8 Jul 2024).
• Limitation—Scope of Rules: Performance and physical validity are only as strong as the embedded rules or equations; deviations from the encoded physics due to unmodeled effects or uncertainties can constrain model flexibility (Faroughi et al., 2022). In LLM-guided scenarios, the quality and comprehensiveness of the rule base directly affect reliability (Gupta et al., 3 Sep 2024).

5. Comparative Perspectives with Alternative Physics-Integrated Neural Approaches

Three primary paradigms are distinguished (Faroughi et al., 2022):

• Physics-Guided NN (PgNN): Training on physics-consistent data, with indirect guidance (no explicit constraints).
• Physics-Informed NN (PiNN): Directly penalize governing law residuals in the loss function.
• Physics-Encoded NN (PeNN): Architectures hard-code physical operators (e.g., differential solvers, evolution blocks) as network layers.

PGCNNs, in many instantiations, most closely align with PiNNs and PeNNs, but with explicit use of convolutional architectural priors and domain transformations. They sit intermediate between purely statistical (PgNN) and fully hard-coded (PeNN) models, often embodying beneficial trade-offs in terms of flexibility, data efficiency, and adherence to physical priors.

6. Outlook and Future Research Directions

Key future challenges for PGCNN research include:

• Unified architectural frameworks: Automating the translation of physical constraints into convolutional and feature map operations for arbitrary domains.
• Rule formalization and explainability: Dynamic, interpretable, and updatable rule modules—potentially sourced from LLMs or expert systems—offer substantial promise for generalization but require rigorous validation against scientific standards (Gupta et al., 3 Sep 2024).
• Robustness to physical model error: Developing strategies for uncertainty quantification when embedded physics is approximate or partially incorrect.
• Integration with traditional solvers: Hybrid models that accelerate or augment established simulation pipelines, for example by serving as surrogates in expensive optimization or risk assessment tasks (Faroughi et al., 2022).
• Expansion to multi-modal, multi-physics settings: Extending methods to capture coupled dynamics and leverage complementary observation modalities.

PGCNNs occupy a central position at the intersection of machine learning and scientific computing, providing a framework for integrating theoretical, empirical, and computational knowledge in domains where data alone is insufficient for reliable discovery or prediction.