Physics-Guided Deep Learning

• Physics-guided deep learning is a hybrid paradigm that combines physical laws and deep neural networks to ensure model predictions adhere to conservation laws and governing equations.
• It employs physics-guided loss functions, architecture strategies, and hybrid simulator models to enforce physical consistency and improve performance with limited or noisy data.
• Applications in medical imaging, fluid dynamics, and weather forecasting illustrate its benefits such as increased sample efficiency, robust generalization, and faster simulation times.

Physics-guided deep learning (PGDL) is the family of methods that integrates prior physical knowledge with deep neural networks so that learning is constrained not only by data but also by governing equations, conservation laws, variational principles, symmetries, geometric structure, or simulator-derived forward models. In the formulation emphasized for dynamical systems, the target state or field $\mathbf{u}$ satisfies a differential relation of the form $$\mathcal{F}(D^k\mathbf{u}(x),D^{k-1}\mathbf{u}(x),\ldots,D\mathbf{u}(x),\mathbf{u}(x),x)=0$$, and deep learning is used either to approximate $\mathbf{u}$, to learn missing components of $\mathcal{F}$, or to accelerate inference while preserving physical structure (Wang et al., 2021). In the data-scarcity literature, PGDL is defined more broadly as deep learning guided by conservation laws, governing equations, or empirical mathematical models, with the central claim that the additional information supplied by physics improves accuracy and generalization when labeled data are scarce, noisy, incomplete, or unavailable (Bai et al., 2022).

1. Definition, scope, and historical positioning

PGDL emerged from the mismatch between two dominant scientific-computing paradigms. Traditional physics-based simulation is interpretable and sample efficient, but it is often computationally intensive, dependent on rigid assumptions, and incomplete when governing laws or closure terms are only partially known. Pure deep learning can emulate nonlinear mappings efficiently, but without explicit physical structure it may violate governing laws and generalize poorly across systems, parameter regimes, or transformed frames (Wang et al., 2021). PGDL addresses this by constraining the hypothesis space with physical structure while retaining the expressive flexibility of neural networks.

In the surveyed literature, PGDL is not a single method class. The field encompasses direct PDE-constrained learning, algorithm-unrolled inverse solvers, hybrid models that combine mechanistic simulators with learned residuals, symmetry-aware architectures, and physically parameterized surrogates. A review focused on data scarcity treats the canonical pattern as a composite objective of the form $$\mathcal{L}=\mathcal{L}_{\text{data}}+\lambda \mathcal{L}_{\text{physics}}$$, but the broader survey emphasizes that physics can also enter through model architecture, learned operators, latent dynamics, or simulator coupling rather than loss terms alone (Bai et al., 2022).

A recurrent misconception is that PGDL is synonymous with physics-informed neural networks. The survey on dynamical systems explicitly organizes the area into four categories—physics-guided loss functions and regularization, physics-guided architecture design, hybrid physics-DL models, and invariant/equivariant models—showing that residual-based PINN formulations are only one branch of the field (Wang et al., 2021).

2. Major mechanisms for injecting physics

The principal integration mechanisms can be organized as follows.

Mechanism	Core idea	Representative papers
Physics-guided loss	Add residual, conservation, or latent-state penalties to data loss	(Bai et al., 2022, Liu et al., 2021, Karami et al., 2023)
Physics-guided architecture	Encode physics in unrolled solvers, coordinate systems, or operator structure	(Yaman et al., 2020, Yue et al., 2024, Gebre et al., 2 Jun 2026)
Hybrid simulator-learning models	Couple learned components with a forward model or surrogate physics engine	(Achlatis et al., 23 Jun 2025, Luo et al., 20 May 2025)
Variational/physical substrate learning	Use energy or action principles to define inference and learning	(Scellier, 2021)

Loss-level integration is the most direct. In the survey formulation, training minimizes a data term plus a residual term, $$\min_\theta \mathcal{L}(\mathbf{u})+\lambda_{\mathcal F}\mathcal{L}_{\mathcal F}(\mathbf{u})$$, with extensions for boundary and initial conditions (Wang et al., 2021). Concrete instances include mass-conservation penalties for viscoelastic tissue deformation, near-field electromagnetic consistency losses for inverse scattering, and thermodynamic or energy-balance penalties in heat-pump stress detection (Karami et al., 2023).

Architecture-level integration hard-codes physical structure into the network itself. In MRI reconstruction, algorithm unrolling turns alternating regularization and data-consistency updates into a trainable 10-iteration network whose data-consistency units explicitly contain the encoding operator ${\bf E}_{\Omega}$ and are solved with conjugate gradient, rather than forcing a CNN to infer the acquisition process implicitly (Yaman et al., 2020). In ptychography, the polar coordinate attention mechanism replaces Euclidean spatial priors with radial-angular correlations aligned with reciprocal-space diffraction physics (Yue et al., 2024). In flood prediction, a hybrid UNet plus Fourier Neural Operator is coupled with shallow-water residuals so that local inundation geometry and basin-scale hydraulic interactions are handled by different architectural components (Gebre et al., 2 Jun 2026).

Hybrid models couple neural networks with simulators or surrogate forward models. Radiotherapy planning uses a two-stage pipeline in which predicted VMAT plan parameters are passed through a frozen differentiable dose predictor, and the planner is optimized in both machine-parameter space and dose space (Achlatis et al., 23 Jun 2025). Weather downscaling and forecasting uses a coordinate-based neural field $f_\theta(x,y,t)$, a candidate PDE library $\Phi(\mathbf{u})\Xi$, and a latent force model $Q_\pi$ so that explicit resolvable meteorological dynamics and unresolved forcings coexist in a single residual (Luo et al., 20 May 2025).

A more foundational variant appears in equilibrium propagation. There, the model itself is a physical or variational system whose state satisfies stationarity conditions, and the gradient is obtained from the sensitivity of local energy derivatives between free and nudged phases: $$\mathcal{F}(D^k\mathbf{u}(x),D^{k-1}\mathbf{u}(x),\ldots,D\mathbf{u}(x),\mathbf{u}(x),x)=0$$0 (Scellier, 2021). This broadens PGDL beyond “physics in the loss” toward “physics as the computational substrate.”

3. Canonical formulations

A general PGDL problem starts from a physical state equation and augments it with learning. In the dynamical-systems survey, the governing relation is written as

$$\mathcal{F}(D^k\mathbf{u}(x),D^{k-1}\mathbf{u}(x),\ldots,D\mathbf{u}(x),\mathbf{u}(x),x)=0$$1

with learning tasks ranging from solving the equation to learning residual dynamics, forecasting future states, or discovering governing equations from data (Wang et al., 2021).

In inverse problems, PGDL often alternates a physics-based consistency step with a learned prior. The MRI formulation is representative: $$\mathcal{F}(D^k\mathbf{u}(x),D^{k-1}\mathbf{u}(x),\ldots,D\mathbf{u}(x),\mathbf{u}(x),x)=0$$2 where $$\mathcal{F}(D^k\mathbf{u}(x),D^{k-1}\mathbf{u}(x),\ldots,D\mathbf{u}(x),\mathbf{u}(x),x)=0$$3 includes coil sensitivities, Fourier encoding, and undersampling, and $$\mathcal{F}(D^k\mathbf{u}(x),D^{k-1}\mathbf{u}(x),\ldots,D\mathbf{u}(x),\mathbf{u}(x),x)=0$$4 is replaced by a learned CNN prior in the unrolled network (Yaman et al., 2020). The same paper shows that even when the architecture is already physics-based, training can still be regularized by randomizing which acquired measurements are enforced in the data-consistency blocks.

In scientific surrogate modeling, conservation laws are commonly imposed as soft penalties. For viscoelastic tissue simulation, the training loss is

$$\mathcal{F}(D^k\mathbf{u}(x),D^{k-1}\mathbf{u}(x),\ldots,D\mathbf{u}(x),\mathbf{u}(x),x)=0$$5

with

$$\mathcal{F}(D^k\mathbf{u}(x),D^{k-1}\mathbf{u}(x),\ldots,D\mathbf{u}(x),\mathbf{u}(x),x)=0$$6

so predicted deformations are penalized when they violate approximate volume preservation (Karami et al., 2023). In inverse scattering, the loss acts not on the final contrast image alone but on hidden electromagnetic state variables such as induced currents and scattered near fields inside the domain of interest, thereby enforcing multiple-scattering consistency (Liu et al., 2021).

Under partial physical knowledge, the residual can itself be parameterized. In weather downscaling and forecasting, the learned dynamics take the form

$$\mathcal{F}(D^k\mathbf{u}(x),D^{k-1}\mathbf{u}(x),\ldots,D\mathbf{u}(x),\mathbf{u}(x),x)=0$$7

where $$\mathcal{F}(D^k\mathbf{u}(x),D^{k-1}\mathbf{u}(x),\ldots,D\mathbf{u}(x),\mathbf{u}(x),x)=0$$8 represents explicit PDE terms available from observed variables and $$\mathcal{F}(D^k\mathbf{u}(x),D^{k-1}\mathbf{u}(x),\ldots,D\mathbf{u}(x),\mathbf{u}(x),x)=0$$9 captures unresolved processes such as omitted vertical transport, diffusivity, or source terms (Luo et al., 20 May 2025). This gray-box structure is increasingly characteristic of PGDL in domains where the governing equations are known only partially at the observation level.

4. Representative application areas

Medical imaging and inverse problems have been especially fertile ground. In accelerated multi-coil MRI, multi-mask supervised training improves conventional supervised physics-guided reconstruction by retrospectively masking acquired measurements during training while leaving inference-time architecture unchanged; on knee MRI the best setting, $\mathbf{u}$0, improved median SSIM/PSNR from $\mathbf{u}$1, $\mathbf{u}$2 dB to $\mathbf{u}$3, $\mathbf{u}$4 dB for uniform sampling (Yaman et al., 2020). In electromagnetic inverse scattering, physics-guided current and field losses improve reconstruction differently by regime: the current-guided loss is strongest at 20 dB on digit-like objects, while the field-guided loss is most robust at 5 dB and under SNR mismatch (Liu et al., 2021). In ptychographic imaging, a dual-branch architecture with polar coordinate attention improves preservation of high-frequency diffraction structure and remains robust at low overlap ratios relative to existing end-to-end models (Yue et al., 2024). In high-resolution X-ray imaging for HED and IFE-relevant experiments, artifact suppression is performed by estimating a multiplicative structured feature layer and dividing it out before transmission reconstruction; in synthetic injection tests, mean SSIM improved from $\mathbf{u}$5 to $\mathbf{u}$6 and from $\mathbf{u}$7 to $\mathbf{u}$8, while filament-length RMSPE was $\mathbf{u}$9 versus $\mathcal{F}$0 for Fourier filtering (Lee et al., 2 May 2026).

Scientific simulation and mechanics form a second major cluster. In viscoelastic tissue simulation, adding a mass-conservation term to a CNN-LSTM surrogate improved generalization on unseen simulation cases by $\mathcal{F}$1 to $\mathcal{F}$2, depending on external-force magnitude (Karami et al., 2023). In bubbly two-phase CFD, coarse-mesh predictions are corrected by a deep feedforward network trained on physics-guided local features and multi-fidelity error data, reducing total cost to about $\mathcal{F}$3 of the high-fidelity simulation while removing unphysical near-wall peaks (Bao et al., 2019). In flow-field denoising, a multi-agent PixelRL system guided by the momentum equation, pressure Poisson equation, and boundary conditions reconstructs noisy DNS and PIV fields without target training data and recovers statistics, spectra, POD modes, and DMD eigenstructure with strong fidelity (Yousif et al., 2023). In flood prediction, coupling UNet and FNO with shallow-water residual losses yielded IoU $\mathcal{F}$4, F1 $\mathcal{F}$5, water-depth RMSE $\mathcal{F}$6 m, and flow-velocity RMSE $\mathcal{F}$7 m/s, while removing physics-based regularization increased water-depth RMSE by $\mathcal{F}$8 (Gebre et al., 2 Jun 2026).

Earth-system and weather applications demonstrate PGDL’s utility under sparse, multiresolution observation. For spatiotemporal temperature reconstruction, a CNN is combined with an annual temperature cycle and a linear ERA5-driven term so that the model learns only the residual fine-scale component; on Landsat and GOES-16 datasets, the proposed method consistently outperformed ATC, ATC+ERA5, and a naïve CNN, including a reduction in GOES-16 test MAE from $\mathcal{F}$9 K to $$\mathcal{L}=\mathcal{L}_{\text{data}}+\lambda \mathcal{L}_{\text{physics}}$$0 K on one dataset (Liu et al., 14 Jul 2025). In weather downscaling and forecasting, PhyDL-NWP uses a continuous neural field, automatic differentiation, a PDE library, and latent force parameterization, improving Huadong downscaling RMSE by $$\mathcal{L}=\mathcal{L}_{\text{data}}+\lambda \mathcal{L}_{\text{physics}}$$1 at 2x and $$\mathcal{L}=\mathcal{L}_{\text{data}}+\lambda \mathcal{L}_{\text{physics}}$$2 at 4x relative to the best baseline while adding only 55K parameters and achieving up to 170x faster module-level cost (Luo et al., 20 May 2025).

Healthcare planning, materials, and energy systems illustrate the breadth of the field. In radiotherapy treatment planning, a two-stage planner first predicts MLC and monitor-unit parameters and then incorporates a dose loss from a frozen differentiable RT dose predictor; on 133 prostate cases, the method achieved $$\mathcal{L}=\mathcal{L}_{\text{data}}+\lambda \mathcal{L}_{\text{physics}}$$3 Gy and $$\mathcal{L}=\mathcal{L}_{\text{data}}+\lambda \mathcal{L}_{\text{physics}}$$4 at the PTV, with improved gamma pass rates in high-dose regions (Achlatis et al., 23 Jun 2025). In crystal generation, the Physics Guided Crystal Generative Model embeds space-group symmetry, distance constraints, and crystallographic structure, improving validity by more than $$\mathcal{L}=\mathcal{L}_{\text{data}}+\lambda \mathcal{L}_{\text{physics}}$$5 relative to FTCP and more than $$\mathcal{L}=\mathcal{L}_{\text{data}}+\lambda \mathcal{L}_{\text{physics}}$$6 relative to CubicGAN; of 2,000 generated materials, 1,869 were successfully optimized by DFT, $$\mathcal{L}=\mathcal{L}_{\text{data}}+\lambda \mathcal{L}_{\text{physics}}$$7 had negative formation energy, and $$\mathcal{L}=\mathcal{L}_{\text{data}}+\lambda \mathcal{L}_{\text{physics}}$$8 had energy-above-hull less than $$\mathcal{L}=\mathcal{L}_{\text{data}}+\lambda \mathcal{L}_{\text{physics}}$$9 eV/atom (Zhao et al., 2022). In semiconductor device modeling, a semi-supervised physics-guided neural network pretrains on FET equations and fine-tunes on a small experimental set, reducing labeled-data requirements by more than $$\min_\theta \mathcal{L}(\mathbf{u})+\lambda_{\mathcal F}\mathcal{L}_{\mathcal F}(\mathbf{u})$$0 for similar or better performance than a traditional neural network (Mishra et al., 2021). In heat-pump stress detection, physics guidance is injected through feature selection, target construction, and physics-regularized losses, with the proposed PG-DNN reported at $$\min_\theta \mathcal{L}(\mathbf{u})+\lambda_{\mathcal F}\mathcal{L}_{\mathcal F}(\mathbf{u})$$1 test accuracy and $$\min_\theta \mathcal{L}(\mathbf{u})+\lambda_{\mathcal F}\mathcal{L}_{\mathcal F}(\mathbf{u})$$2 validation accuracy on the When2Heat dataset (Alam et al., 23 Nov 2025).

5. Benefits, misconceptions, and limitations

Across surveys and domain papers, the recurring benefits are improved sample efficiency, physical consistency, interpretability, and generalization. The survey on dynamical systems repeatedly argues that physics-guided models shrink the hypothesis space, produce scientifically valid predictions, and generalize better under transformed or shifted conditions because laws, constraints, and symmetries remain valid beyond the training set (Wang et al., 2021). The review on data scarcity makes the same point more directly: physics laws can provide “naturally annotated” supervision through residuals, boundary conditions, or empirical equations when high-quality labels are limited (Bai et al., 2022).

These gains do not imply that PGDL guarantees physically exact outputs. Many prominent methods use soft constraints rather than hard ones. The viscoelastic tissue surrogate penalizes volume change but does not enforce exact incompressibility, and the radiotherapy planner uses a learned dose predictor rather than an exact transport solver (Karami et al., 2023). Likewise, weather downscaling uses a partial PDE plus latent force parameterization, not the full primitive equations (Luo et al., 20 May 2025). A second misconception is that “more physics” or “more augmentation” is automatically beneficial. In MRI, increasing the number of retrospective masks from $$\min_\theta \mathcal{L}(\mathbf{u})+\lambda_{\mathcal F}\mathcal{L}_{\mathcal F}(\mathbf{u})$$3 to $$\min_\theta \mathcal{L}(\mathbf{u})+\lambda_{\mathcal F}\mathcal{L}_{\mathcal F}(\mathbf{u})$$4 or $$\min_\theta \mathcal{L}(\mathbf{u})+\lambda_{\mathcal F}\mathcal{L}_{\mathcal F}(\mathbf{u})$$5 did not improve performance monotonically (Yaman et al., 2020). In inverse scattering, the induced-current loss improves high-SNR in-distribution performance but can become more training-set dependent and less robust on polygonal objects or under strong noise, whereas the field-guided loss is more stable (Liu et al., 2021).

The main limitations identified across the review literature are also consistent: imbalanced loss terms, optimization difficulty for complex nonlinear physics, singularities and discontinuities, computational cost, imperfect or incomplete physics, and underdeveloped uncertainty quantification (Bai et al., 2022). The survey further notes that hard architectural biases can become overly restrictive when the physical prior is only approximate, while soft penalties may still be violated at test time (Wang et al., 2021).

6. Current directions and open problems

Several trajectories define the present frontier. One is the move from fully known physics to partial-physics gray-box models, where explicit PDE structure is combined with latent parameterization of missing terms, as in weather dynamics (Luo et al., 20 May 2025). A second is differentiable simulator coupling, where a learned planner or reconstructor is optimized through a frozen forward model, as in radiotherapy dose prediction (Achlatis et al., 23 Jun 2025). A third is geometry- and modality-aligned inductive bias, visible in polar-coordinate attention for diffraction data and multi-scale operator designs for hydrodynamics (Yue et al., 2024).

The review literature emphasizes several unresolved needs: better loss balancing, stronger theory for convergence and generalization, more robust handling of discontinuities and multiscale nonlinear physics, principled transfer learning criteria, and improved uncertainty quantification (Bai et al., 2022). Application papers add further requirements: more explicit deliverability constraints in radiotherapy, broader anatomy and protocol coverage, and stronger out-of-distribution safeguards such as the deep-ensemble epistemic uncertainty used in X-ray artifact suppression (Achlatis et al., 23 Jun 2025). More generally, the field is moving toward modular systems in which physical structure may define the loss, the architecture, the representation, the learning substrate, or all four simultaneously.

In that broader sense, physics-guided deep learning is best understood not as a specific algorithmic recipe but as a design philosophy: retain whatever physical structure is reliable, learn only what is unknown or computationally prohibitive, and make the interaction between model and physics explicit enough that accuracy gains do not come at the cost of scientific plausibility.