Physics-Driven Neural Networks (PDNN)

• PDNN is a family of neural networks that integrate physical laws through regularization, inverse-problem solving, and architecture design.
• Methods include using PDE/ODE residuals, differentiable physics solvers, and parameter embedding to guide the learning process.
• Applications span electromagnetics, optical imaging, molecular dynamics, and quantitative MRI, demonstrating improved efficiency and accuracy.

Searching arXiv for recent and foundational papers on Physics-Driven Neural Networks and closely related formulations. Physics-Driven Neural Network (PDNN) denotes a family of neural-network formulations in which physical laws, physical structure, or physical substrates constrain learning. In the literature, the term is used for several related constructions: neural surrogates trained with PDE or ODE residuals, inverse-problem solvers coupled to differentiable forward physics, architectures whose layers explicitly encode mechanistic structure, and even networks whose “neurons” and “synapses” are realized as physical devices. Taken together, these works suggest that PDNN is best treated as an umbrella designation rather than a single canonical model class (Mohammad-Djafari et al., 2024, Wycoff et al., 2022, Lim et al., 2021).

1. Terminological scope and historical development

One important lineage defines PDNNs through physics-driven regularization: the neural approximation $\tilde u(x;\theta)$ is trained not only against observations, but also against a known governing equation, so that divergence from the law is penalized during optimization (Nabian et al., 2018). A closely related formulation appears in general analyses of physics-informed neural networks, where the solution ansatz $u_\theta$ is trained simultaneously on data points and collocation points, with the PDE residual entering the objective explicitly (Small, 2023).

A second lineage uses the term for model-based inverse-problem networks. Mohammad-Djafari et al. classify model-based and physics-informed deep architectures into five classes: explicit analytical-solution networks, transform-domain decomposition networks, operator-decomposition networks, optimization-unfolding networks, and physics-informed neural networks (Mohammad-Djafari et al., 2024). Within this broader perspective, a PDNN need not be a residual-loss PINN; it may instead embed a forward operator, an iterative algorithm, or a known transform.

A third lineage emphasizes physics-guided architecture design. PhyDNN decomposes drag prediction into pressure and shear pathways and supplements supervision with aggregate physical statistics (Muralidhar et al., 2019). Phy-Taylor introduces Taylor-monomial augmentation, noise suppressors, and physics-guided neural-network editing so that known substructures are fixed while only unknown components remain trainable (Mao et al., 2022). Multi-fidelity PIDNNs train first on approximate governing equations and then transfer learned features to scarce high-fidelity data (Chakraborty, 2020).

A fourth lineage treats PDNNs as physical learning machines. In “Desynchronous Learning in a Physics-Driven Learning Network,” the computational graph is an electrical network of adjustable resistors obeying local physics and local update rules (Wycoff et al., 2022). In “Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems,” the layers themselves are optical, mechanical, or electrical systems trained by Physics-Aware Training (Wright et al., 2021).

2. Recurrent mathematical formulations

The most common mathematical pattern augments a data term with a physics term. In physics-driven regularization, the total objective is written as

$$L_{\rm total}(\theta)=L_{\rm data}(\theta)+\lambda L_{\rm phys}(\theta),$$

with

$$L_{\rm data}(\theta)=\frac{1}{N}\sum_{i=1}^N\|y_i-\tilde u(x_i;\theta)\|^2, \qquad L_{\rm phys}(\theta)=\frac{1}{N_r}\sum_{j=1}^{N_r}\bigl\|\mathcal D[x_j,\tilde u(x_j;\theta)]\bigr\|^2.$$

Here $\mathcal D$ is the known differential operator and the derivatives of $\tilde u$ are obtained by automatic differentiation (Nabian et al., 2018).

In PDE-oriented PDNNs, this template is specialized to the governing law of the domain. MaxwellNet, for example, trains a network that predicts the electric field from the relative-permittivity distribution by minimizing the squared residual of the time-harmonic source-free Maxwell curl-curl equation,

$$\nabla\times\nabla\times E(\mathbf r)-k_0^2\,\varepsilon_r(\mathbf r)\,E(\mathbf r)=0,$$

discretized on a Yee grid with a perfectly matched layer (Lim et al., 2021). Similar constructions appear in PINN analyses for wave equations, Burgers’ equation, and seismic imaging, with collocation enforcing the PDE away from the data anchors (Small, 2023).

Inverse-scattering PDNNs replace direct PDE residuals by a differentiable physics loop. The unknown permittivity $\epsilon_r$ is represented by a neural network output, inserted into a Method-of-Moments forward solver, and optimized through a loss containing a data-consistency term, a lower-bound constraint on $\mathrm{Re}\{\hat\epsilon_r\}$, and a total-variation regularizer (Du et al., 22 Jul 2025). The improved IPDNN preserves the same coupling but introduces the GLOW activation and dynamic subregion identification to stabilize convergence and reduce cost (Du et al., 10 Dec 2025).

Not all PDNNs use physics as a residual penalty. In the terahertz holography PDNN, angular spectrum theory is used to generate synthetic diffraction/object pairs, and self-training combines these labeled synthetic data with unlabeled experimental data through pseudo-label refinement (Xiang et al., 2024). In quantitative MRI, TR, TE, and TI are embedded directly into the network so that the model learns parametric dependencies of MR signal formation rather than a fixed image-to-image mapping (Chen et al., 11 Aug 2025).

3. Architectural families and mechanisms of physics insertion

The literature supports several distinct ways of inserting physics into a neural model.

Family	Mechanism	Representative works
Residual-penalty PDNNs	PDE/ODE residual added to loss at collocation points	(Nabian et al., 2018, Small, 2023, Lim et al., 2021)
Forward-solver-coupled PDNNs	Differentiable simulator placed inside training loop	(Du et al., 22 Jul 2025, Du et al., 10 Dec 2025, Razakh et al., 2020)
Multi-fidelity PDNNs	Low-fidelity physics pretraining, then high-fidelity transfer learning	(Chakraborty, 2020)
Physics-compatible architectures	Monomials, masks, editing, suppressors encode known structure	(Mao et al., 2022, Muralidhar et al., 2019)
Physics-generated or parameter-embedded models	Forward physics used for synthetic supervision or acquisition-parameter embedding	(Xiang et al., 2024, Chen et al., 11 Aug 2025)
Physical-substrate PDNNs	Real physical systems act as layers or synapses	(Wright et al., 2021, Wycoff et al., 2022)

Within these families, the architectural choices differ substantially. Phy-Taylor constructs the augmented feature vector $\mathfrak m(\mathbf z,r)$ from Taylor monomials up to order $u_\theta$0, splits each layer into a knowledge matrix for known coefficients and an uncertainty matrix for learned terms, and uses binary masks to freeze or cut links that are already fixed by physics (Mao et al., 2022). This is a strong architectural prior rather than a mere regularizer.

Other works retain conventional deep backbones but insert physics-driven modules. PPRNet is a multi-scale U-Net–style phase-retrieval network whose Hybrid Unwinding Blocks impose measured Fourier magnitudes at each scale; the network remains feedforward and end-to-end trainable, while the forward pass is guided by the measurement model rather than by a separate iterative reconstruction loop (Ye et al., 2022). MaxwellNet for nonlinear optical scattering adds a small fully connected subnetwork that maps incident intensity to adjustments of a convolutional block, thereby tuning the network to the Kerr-effect regime without retraining from scratch (Gigli et al., 2022).

This variety is consistent with the five-class taxonomy of model-based and physics-informed structures. It also shows that “physics-driven” does not imply a single depth, layer type, or optimizer; the defining feature is the way physical prior information constrains representation or learning (Mohammad-Djafari et al., 2024).

4. Distributed learning networks and physical realizations

In the resistor-network PDNN, nodes are electrical junctions with voltages $u_\theta$1, and synapses are adjustable resistors $u_\theta$2 whose conductances $u_\theta$3 play the role of weights. Learning alternates between a free phase, where input voltages are applied and the network relaxes to $u_\theta$4, and a clamped phase, where output nodes are nudged toward desired values and the network re-equilibrates to $u_\theta$5. Each resistor updates using only its own free-phase and clamped-phase voltage drops:

$u_\theta$6

Desynchronous learning is implemented by updating each resistor only with probability $u_\theta$7. In a continuous-resistance simulation with 143 edges and a 2-in/2-out regression task, test-error curves collapse when plotted versus the effective step $u_\theta$8, showing no loss in learning rate for $u_\theta$9. In a 16-edge discrete-resistor experiment on allosteric tasks, desynchronization lowers the final test error by up to $$L_{\rm total}(\theta)=L_{\rm data}(\theta)+\lambda L_{\rm phys}(\theta),$$0 compared with $$L_{\rm total}(\theta)=L_{\rm data}(\theta)+\lambda L_{\rm phys}(\theta),$$1, and the system functions equally well for $$L_{\rm total}(\theta)=L_{\rm data}(\theta)+\lambda L_{\rm phys}(\theta),$$2 as low as $$L_{\rm total}(\theta)=L_{\rm data}(\theta)+\lambda L_{\rm phys}(\theta),$$3 (Wycoff et al., 2022).

A different notion of physical PDNN uses real physical systems as trainable layers. In the deep physical neural network framework, a depth-$$L_{\rm total}(\theta)=L_{\rm data}(\theta)+\lambda L_{\rm phys}(\theta),$$4 model is a cascade

$$L_{\rm total}(\theta)=L_{\rm data}(\theta)+\lambda L_{\rm phys}(\theta),$$5

where each $$L_{\rm total}(\theta)=L_{\rm data}(\theta)+\lambda L_{\rm phys}(\theta),$$6 is a controllable physical process. Physics-Aware Training performs the forward pass on the real device and the backward pass through a differentiable surrogate. The paper reports three implementations: an optical network based on second-harmonic generation achieving 93% test accuracy on 7 spoken vowels, a mechanical network based on multimode plate oscillations achieving 87% test accuracy on MNIST, and an electronic network based on a nonlinear transistor oscillator achieving 93% test accuracy on MNIST. The reported self-simulation analysis estimates speed-ups of $$L_{\rm total}(\theta)=L_{\rm data}(\theta)+\lambda L_{\rm phys}(\theta),$$7–$$L_{\rm total}(\theta)=L_{\rm data}(\theta)+\lambda L_{\rm phys}(\theta),$$8 and energy savings of $$L_{\rm total}(\theta)=L_{\rm data}(\theta)+\lambda L_{\rm phys}(\theta),$$9–$$L_{\rm data}(\theta)=\frac{1}{N}\sum_{i=1}^N\|y_i-\tilde u(x_i;\theta)\|^2, \qquad L_{\rm phys}(\theta)=\frac{1}{N_r}\sum_{j=1}^{N_r}\bigl\|\mathcal D[x_j,\tilde u(x_j;\theta)]\bigr\|^2.$$0 relative to digital simulation (Wright et al., 2021).

These two strands illustrate a substantive distinction inside the PDNN literature. In one case, physics constrains optimization over software parameters; in the other, physics is itself the trainable computational medium.

5. Representative application domains

Electromagnetics is one of the most developed PDNN domains. MaxwellNet trains without precomputed field labels by minimizing the residual of Maxwell’s equations and, on a held-out set of 20 new aspheric lens shapes, reports mean relative errors of $$L_{\rm data}(\theta)=\frac{1}{N}\sum_{i=1}^N\|y_i-\tilde u(x_i;\theta)\|^2, \qquad L_{\rm phys}(\theta)=\frac{1}{N_r}\sum_{j=1}^{N_r}\bigl\|\mathcal D[x_j,\tilde u(x_j;\theta)]\bigr\|^2.$$1 for TE and $$L_{\rm data}(\theta)=\frac{1}{N}\sum_{i=1}^N\|y_i-\tilde u(x_i;\theta)\|^2, \qquad L_{\rm phys}(\theta)=\frac{1}{N_r}\sum_{j=1}^{N_r}\bigl\|\mathcal D[x_j,\tilde u(x_j;\theta)]\bigr\|^2.$$2 for TM, with inference time below 10 ms per TE-mode evaluation on a GTX 1070, compared with about 1 s for FDTD (Lim et al., 2021). Its nonlinear extension for Kerr media reports average relative errors of approximately 1.7–3.0% in TE and approximately 2.5–3.9% in TM, with inference times of about 15 ms and 21 ms, respectively, versus 5–20 s for COMSOL (Gigli et al., 2022). For global photonic design, GLOnet reframes adjoint optimization as training a conditional generative network; across $$L_{\rm data}(\theta)=\frac{1}{N}\sum_{i=1}^N\|y_i-\tilde u(x_i;\theta)\|^2, \qquad L_{\rm phys}(\theta)=\frac{1}{N_r}\sum_{j=1}^{N_r}\bigl\|\mathcal D[x_j,\tilde u(x_j;\theta)]\bigr\|^2.$$3 and $$L_{\rm data}(\theta)=\frac{1}{N}\sum_{i=1}^N\|y_i-\tilde u(x_i;\theta)\|^2, \qquad L_{\rm phys}(\theta)=\frac{1}{N_r}\sum_{j=1}^{N_r}\bigl\|\mathcal D[x_j,\tilde u(x_j;\theta)]\bigr\|^2.$$4, its top-1 samples matched or exceeded the best adjoint-restart design in 75% of cases and were within 5% in 92% of cases, while using approximately 10 times fewer FEM/RCWA solves per optimized design ensemble (Jiang et al., 2019).

Electromagnetic inverse scattering is another major testbed. The per-instance PDNN solver for 2-D TM inverse scattering combines a CNN with a Method-of-Moments forward model and reports relative errors of 1.5–6% on canonical targets, with runtime over 3000 iterations dropping from about 362 s to 48–85 s after subregion cropping (Du et al., 22 Jul 2025). The improved IPDNN introduces the Gaussian-Localized Oscillation-Suppressing Window (GLOW), dynamic scatter subregion identification, and transfer learning; GLOW converges in roughly 500 iterations to 2–3% error while alternative activations stagnate above 5–10%, dynamic subregions reduce CPU/GPU time by 30–50% with no loss of accuracy, and transfer learning accelerates convergence in defect-detection tasks by a factor of 5, from about 200 iterations to about 40 (Du et al., 10 Dec 2025).

Optical imaging and phase retrieval provide further variants. In terahertz holography, a physics-driven self-training neural network uses angular spectrum theory to generate 30,000 synthetic diffraction/object pairs and combines them with about 22 real experimental diffraction patterns, augmented to about 66 by added noise; the reported blind-test MSEs on unseen digit pairs are approximately 0.0043 for the first object and 0.0041 for the second, and experimental reconstructions are described as near-ghost-free even when only amplitude diffraction is used at inference (Xiang et al., 2024). For single-shot phase retrieval, PPRNet uses a three-scale U-Net-style backbone with multi-scale physics-driven unwinding blocks, about 19.8M parameters, about 35G MACs, and 4–6 ms inference on a single RTX-3090; on simulated 12-bit defocused data it achieves 32.99 dB and 34.18 dB PSNR for magnitude and phase on the correlated dataset, and on real COCO measurements it reaches about 17.5 dB PSNR, outperforming HIO, PrDeep, and HIO-UNet (Ye et al., 2022).

Molecular and mechanical dynamics form another cluster. The PND software treats molecular dynamics as a physics-informed trajectory-learning problem: an MLP maps time to concatenated positions and velocities, and the loss combines residual, initial/boundary, energy, and momentum terms. In the Argon FCC example, it reports energy conservation within $$L_{\rm data}(\theta)=\frac{1}{N}\sum_{i=1}^N\|y_i-\tilde u(x_i;\theta)\|^2, \qquad L_{\rm phys}(\theta)=\frac{1}{N_r}\sum_{j=1}^{N_r}\bigl\|\mathcal D[x_j,\tilde u(x_j;\theta)]\bigr\|^2.$$5 in LJ units over 25 ps, trajectory RMSD below $$L_{\rm data}(\theta)=\frac{1}{N}\sum_{i=1}^N\|y_i-\tilde u(x_i;\theta)\|^2, \qquad L_{\rm phys}(\theta)=\frac{1}{N_r}\sum_{j=1}^{N_r}\bigl\|\mathcal D[x_j,\tilde u(x_j;\theta)]\bigr\|^2.$$6 for randomly selected atoms, and wall-clock training times that scale linearly with $$L_{\rm data}(\theta)=\frac{1}{N}\sum_{i=1}^N\|y_i-\tilde u(x_i;\theta)\|^2, \qquad L_{\rm phys}(\theta)=\frac{1}{N_r}\sum_{j=1}^{N_r}\bigl\|\mathcal D[x_j,\tilde u(x_j;\theta)]\bigr\|^2.$$7 and $$L_{\rm data}(\theta)=\frac{1}{N}\sum_{i=1}^N\|y_i-\tilde u(x_i;\theta)\|^2, \qquad L_{\rm phys}(\theta)=\frac{1}{N_r}\sum_{j=1}^{N_r}\bigl\|\mathcal D[x_j,\tilde u(x_j;\theta)]\bigr\|^2.$$8 on an HPC node (Razakh et al., 2020). PINODE embeds Lagrangian mechanics directly into a neural ODE, modeling only the non-conservative forces with a small MLP; for a real cart-pole system trained from about 8 minutes of random control data, the reported error histograms over 1000 runs show mean and standard deviation 2–5 times smaller than those of a pure ODE model in all four state dimensions (Roehrl et al., 2020).

Quantitative MRI illustrates parameter-embedded PDNN design. PDPE-Net takes T1-weighted, T2-weighted, and T2-FLAIR images as input and embeds TR, TE, and TI through a learnable parameter-embedding module. On internal and external test sets it reports PSNR values exceeding 34 dB and SSIM values above 0.92 for all synthesized T1, T2, and PD maps, and the model is described as accurately synthesizing maps for unseen pathological regions (Chen et al., 11 Aug 2025).

6. Limitations, misconceptions, and current trajectories

A common misconception is that PDNN is synonymous with PINN. The cited literature does not support that restriction. Some PDNNs are classical PINNs with residual losses; others are model-based inverse solvers, physics-compatible architectures, transfer-learning systems, or physical devices trained in situ (Mohammad-Djafari et al., 2024, Wright et al., 2021, Wycoff et al., 2022). A second misconception is that physics must appear as a hard constraint. In practice, physics may enter as a soft residual penalty, a differentiable forward solver, synthetic supervision generated from a known operator, parameter embedding, architectural masking, or a local update rule (Nabian et al., 2018, Xiang et al., 2024, Chen et al., 11 Aug 2025, Mao et al., 2022).

The limitations are similarly heterogeneous. General PINN analyses report that training can stall in local minima, that accuracy deteriorates far from data anchors or where PDE coefficients are discontinuous, and that the choice of architecture, $$L_{\rm data}(\theta)=\frac{1}{N}\sum_{i=1}^N\|y_i-\tilde u(x_i;\theta)\|^2, \qquad L_{\rm phys}(\theta)=\frac{1}{N_r}\sum_{j=1}^{N_r}\bigl\|\mathcal D[x_j,\tilde u(x_j;\theta)]\bigr\|^2.$$9, sampling budget, and optimizer is often problem-specific (Small, 2023). Physics-driven regularization increases training cost when high-order derivatives must be differentiated and may require separate prediction of all PDE variables to penalize the intended laws (Nabian et al., 2018). Per-instance inverse-scattering PDNNs remain slower than one-shot data-driven inference because the scattered field must be recomputed at every iteration, and they can become unstable below about 5 dB SNR (Du et al., 22 Jul 2025). Physics-Aware Training relies on a surrogate $\mathcal D$0 that approximates the true physical module well enough for surrogate gradients to provide descent directions for the real forward loss (Wright et al., 2021).

Current trajectories in the literature point toward further hybridization. Model-based surveys explicitly recommend combining explicit $\mathcal D$1-layers, unrolled algorithms, and PDE-residual units inside single systems (Mohammad-Djafari et al., 2024). The decentralized resistor-network line proposes nonlinear extensions using transistors or memristors in place of resistors and highlights hardware candidates such as 3D nanofabricated resistor nets, memristor arrays, and fluidic-mechanical networks (Wycoff et al., 2022). A more recent spiking extension, NeuroPINNs, replaces standard activations with Variable Spiking Neurons and uses stochastic projection to form PDE losses; on benchmark PDEs it reports synaptic-energy ratios of approximately 0.29–0.58 relative to continuous PINNs, while remaining within 1%–3% relative $\mathcal D$2 error of continuous baselines on several tasks (Garg et al., 8 Nov 2025).

Across these directions, the persistent theme is not a single architecture but a methodological commitment: the model is shaped by known physics at the level of representation, optimization, or hardware. That breadth explains both the productivity and the terminological looseness of the PDNN literature.