Physics-Integrated CNNs (PInteCNNs)

• PInteCNNs are convolutional models that embed physical laws directly into learning via fixed differential kernels, residual terms, and latent physics modules.
• They integrate diverse techniques—from operator-based filters to nonlocal PIDE blocks—that enforce boundary conditions and governing equations during forward propagation.
• Applications range from solving PDEs on regular grids to enhancing image-based inference tasks, demonstrating versatility in improving numerical accuracy and interpretability.

Searching arXiv for the cited papers to ground the article in the current literature. Physics-Integrated Convolutional Neural Networks (PInteCNNs) are convolutional models in which physically meaningful structure is inserted directly into learning, inference, or representation. In the PDE literature, this often denotes CNNs operating on regular grids while minimizing residuals assembled from discrete differential operators so that the predicted fields satisfy governing equations and boundary conditions; in the irregular-domain mapping literature, one paper explicitly states that PICNN and PInteCNN refer to the same concept in that setting (Zhou et al., 10 Jun 2025). Across the broader literature, however, the term covers a wider family: PDE- and PIDE-derived update rules, fixed finite-difference kernels, latent-space physics modules, recurrent cells built from discretized dynamics, rule-guided layers, and physics-aware preprocessing that exposes invariant or bifurcation-relevant signal structure to a CNN (Bohn et al., 2021).

1. Terminology and conceptual scope

The terminology is not uniform. The literature uses PICNN, PICN, PD-CNN, PGCNN, PhICNet, and PhysNet for related but non-identical constructions. In one line of work, PICNN denotes a CNN trained on a rectangular, image-like grid with physics-based losses assembled from discrete differential operators, typically finite differences, so that the output satisfies a PDE and its boundary conditions (Zhou et al., 10 Jun 2025). In another, PICN is a shallow deconvolution-convolution solver that generates a field on a grid and then applies pre-trained finite-difference filters to form “difference fields” used in the physics-informed loss (Shi et al., 2022).

This diversity matters because PInteCNNs are not restricted to the standard PINN template. A common misconception is that physics integration means only adding PDE residual terms to the final output. The literature is broader. Some models place the governing law in the loss, as in CNN-based solvers for Laplace, Poisson, Darcy, or Navier–Stokes systems. Others embed operators directly in the forward propagation, as in PIDE-inspired nonlocal blocks. Still others impose physics or domain knowledge at the level of latent features, recurrent state evolution, object-detection logits, or input preprocessing (Bohn et al., 2021).

The distinction from adjacent categories is therefore structural. Relative to fully connected PINNs, PInteCNNs exploit locality, weight sharing, and tensor-friendly field representations on grids (Shi et al., 2022). Relative to broader physics-guided neural networks, some PInteCNNs are CNN-specific and task-specific: one detection framework, for example, inserts custom rule layers that act on boxes and logits rather than on PDE fields (Gupta et al., 2024). Relative to output-level physics-informed CNNs, latent-physics models place the governing dynamics inside intermediate feature maps rather than only at the network head (Das et al., 30 Mar 2026).

2. Modes of physics integration in convolutional architectures

One major design pattern is operator integration through fixed or semi-fixed convolutional kernels. In the shallow PICN formulation, a deconvolution layer generates a grid field $\hat{u}$, and a convolution layer with pre-trained filters implements discrete operators such as the Laplacian and first derivatives. The canonical $3\times3$ Laplacian kernel is

$$W_\Delta=\frac{1}{h^2} \begin{bmatrix} 0 & 1 & 0\ 1 & -4 & 1\ 0 & 1 & 0 \end{bmatrix},$$

and analogous central-difference kernels are used for $\partial_x$ and $\partial_y$ (Shi et al., 2022). This construction makes the physical operator an explicit part of the architecture rather than an implicit by-product of automatic differentiation.

A second pattern is forward-propagation derived from physical evolution equations. In "Deep Neural Networks and PIDE discretizations," nonlocal CNN blocks are designed as residual updates corresponding to discretized PIDEs. The central operator has the form

$$(\mathcal{K}u)(x)=\int_{\Omega} K(x,y)\big(u(y)-u(x)\big)\,dy,$$

and the implemented blocks use two-stage, Verlet-like residual updates with learned $1\times1$ channel mixers and feature-dependent affinities. This line treats long-range diffusion, fractional Laplacians, and inverse fractional Laplacians as model components rather than merely as training constraints (Bohn et al., 2021).

A third pattern is physics-inspired remodeling of the basic convolutional unit. The IC-network literature derives an “Inter-layer Collision” structure from elastic-collision analogies. Its convolutional form replaces a standard response $w_i * X$ with

$$u_i = w_i * X + \sigma\!\big(w_i * X - (I ** X) w'_i\big),$$

where $I ** X$ is a depthwise separable convolution with all-one kernels and $3\times3$0 is a learned channel vector (An et al., 2019). Here the integration is architectural rather than residual-based: no PDE loss is imposed, but a physics-inspired interaction law changes the representational primitive itself.

A fourth pattern is multi-scale receptive-field integration matched to numerical discretization. MRF-PINN uses six parallel encoder-decoder branches with adaptive dilations

$3\times3$1

so that receptive fields remain proportional to the grid scale as resolution changes. The branch outputs are fused by a learned linear combination, and PDE residuals are evaluated with high-order finite-difference kernels implemented as convolutions (Zhang et al., 2022).

3. PDE-solving formulations on regular grids

The most established PInteCNN usage remains the solution of PDEs on Cartesian grids. In the steady-flow setting, a physics-driven CNN can be trained without labels by minimizing residuals of the incompressible Navier–Stokes equations and boundary conditions. One U-Net-based formulation outputs $3\times3$2 and computes derivatives through fixed $3\times3$3 filters such as

$3\times3$4

That work reports that “a first-order accuracy has been achieved,” and for cylinder flow it learns both creeping and separated steady regimes, including adhered twin vortices (Ma et al., 2021).

MRF-PINN extends this regular-grid strategy by combining multi-receptive-field CNN branches with high-order finite differences and Taylor-polynomial ghost-node padding at boundaries. Its total loss uses weighted PDE, boundary, initial, and data terms,

$3\times3$5

and, for the axisymmetric Navier–Stokes case, the paper sets

$3\times3$6

through a dimensional-balance argument rather than dynamic reweighting (Zhang et al., 2022). This suggests a design principle that is specific to convolutional PDE solvers: the numerical discretization, the receptive field, and the loss scaling are co-dependent choices rather than separable hyperparameters.

Auto-PICNN systematizes that principle. It treats both the loss design and the CNN architecture as search spaces. Stage 1 uses Bayesian optimization to search residual norms, derivative kernels, boundary treatments, residual weighting operators, and related loss factors; Stage 2 uses reinforcement-learning-based architecture search over U-Net-style or simple stacked CNN spaces (Zhou et al., 2024). On the reported test sets, the automatically searched models outperform manually designed baselines across heat, Poisson, Darcy, and steady Navier–Stokes benchmarks, including a velocity and pressure test error of $3\times3$7 and $3\times3$8 on the Navier–Stokes case versus $3\times3$9 and $$W_\Delta=\frac{1}{h^2} \begin{bmatrix} 0 & 1 & 0\ 1 & -4 & 1\ 0 & 1 & 0 \end{bmatrix},$$0 for the baseline (Zhou et al., 2024).

4. Geometry, mappings, and manifolds

A central obstacle for convolutional PDE solvers is geometric irregularity. One recent solution is to learn an irregular-to-regular mapping with a PINN and then run the CNN solver on the mapped rectangular domain. The mapping $$W_\Delta=\frac{1}{h^2} \begin{bmatrix} 0 & 1 & 0\ 1 & -4 & 1\ 0 & 1 & 0 \end{bmatrix},$$1 takes computational coordinates $$W_\Delta=\frac{1}{h^2} \begin{bmatrix} 0 & 1 & 0\ 1 & -4 & 1\ 0 & 1 & 0 \end{bmatrix},$$2 to physical coordinates $$W_\Delta=\frac{1}{h^2} \begin{bmatrix} 0 & 1 & 0\ 1 & -4 & 1\ 0 & 1 & 0 \end{bmatrix},$$3, and is trained with

$$W_\Delta=\frac{1}{h^2} \begin{bmatrix} 0 & 1 & 0\ 1 & -4 & 1\ 0 & 1 & 0 \end{bmatrix},$$4

where $$W_\Delta=\frac{1}{h^2} \begin{bmatrix} 0 & 1 & 0\ 1 & -4 & 1\ 0 & 1 & 0 \end{bmatrix},$$5 imposes a harmonic-map-like smoothness, $$W_\Delta=\frac{1}{h^2} \begin{bmatrix} 0 & 1 & 0\ 1 & -4 & 1\ 0 & 1 & 0 \end{bmatrix},$$6 aligns mapped boundary points, and $$W_\Delta=\frac{1}{h^2} \begin{bmatrix} 0 & 1 & 0\ 1 & -4 & 1\ 0 & 1 & 0 \end{bmatrix},$$7 penalizes negative Jacobian determinant to prevent folding (Zhou et al., 10 Jun 2025). After training, structured grids are obtained by mapping the rectangular nodes back into the irregular physical domain, and the CNN or a classical finite-difference method operates on the regular computational grid using mapping-induced Jacobian weights.

For the Laplace equation in that mapped domain, the residual is written in a compact Jacobian-weighted form,

$$W_\Delta=\frac{1}{h^2} \begin{bmatrix} 0 & 1 & 0\ 1 & -4 & 1\ 0 & 1 & 0 \end{bmatrix},$$8

with Dirichlet values imposed directly on the rectangular edges (Zhou et al., 10 Jun 2025). The reported evaluation is qualitative rather than metric-driven: heat-conduction solutions on irregular domains show good agreement with OpenFOAM, and PICNN results on PINN-generated grids are visually comparable to PICNN on FDM-generated grids and to OpenFOAM for the illustrated Navier–Stokes benchmark (Zhou et al., 10 Jun 2025).

On manifolds, the geometry is intrinsic rather than mapped. For PDEs on the sphere, PICNNs are analyzed through spherical harmonics, Sobolev spaces, and localized complexity. The governing operator includes the Laplace–Beltrami operator, and the physics-informed loss is the mean-squared residual over $$W_\Delta=\frac{1}{h^2} \begin{bmatrix} 0 & 1 & 0\ 1 & -4 & 1\ 0 & 1 & 0 \end{bmatrix},$$9, with no boundary term because the sphere is boundaryless (Lei et al., 2023). The paper proves a Sobolev-norm approximation bound,

$\partial_x$0

and derives convergence rates in which the intrinsic manifold dimension $\partial_x$1, rather than the ambient dimension $\partial_x$2, controls the polynomial exponent (Lei et al., 2023). This line anchors PInteCNNs not only as engineering solvers but also as objects of approximation theory.

5. Beyond output-level PDE residuals

The literature also includes PInteCNNs in which physics is not attached only to the final prediction. PhysNet embeds a Fisher–KPP reaction–diffusion model of tumor growth inside intermediate ResNet-50 features. From the layer-3 feature map $\partial_x$3, two heads predict a latent tumor density field $\partial_x$4 and its temporal derivative $\partial_x$5, and a fixed Laplacian kernel computes

$\partial_x$6

The total loss combines classification, physics residual, boundary smoothness, and pseudo-time consistency, with an EMA-based schedule for $\partial_x$7 (Das et al., 30 Mar 2026). This model reports $\partial_x$8 accuracy and $\partial_x$9 F1 on a $\partial_y$0-image brain MRI dataset, while adding approximately $\partial_y$1M parameters and about $\partial_y$2 inference overhead relative to vanilla ResNet-50 (Das et al., 30 Mar 2026).

PhICNet shows another extension: recurrent rather than static integration. It builds a convolutional recurrent network whose cell state encodes the temporal order of the PDE and whose spatial operators are fixed convolutional approximations of differential terms. For an inhomogeneous PDE

$\partial_y$3

the model separates the homogeneous dynamics from a latent, time-varying source term and predicts

$\partial_y$4

where $\partial_y$5 is a residual encoder-decoder for source evolution (Saha et al., 2020). In the reported heat, wave, and Burgers experiments, the framework outperforms PDE-RNN+CNN, ConvLSTM, and RED-Net on long-horizon forecasting and source identification (Saha et al., 2020).

The same broadened conception appears in object detection and dynamical-systems monitoring. PGCNN augments Faster R-CNN with rule-guided layers such as Redundancy Elimination, CAWAL, HWAD, and a Shape-Based layer; here the “physics” consists of contextual, geometric, and common-sense rules encoded as differentiable operators on logits or detections (Gupta et al., 2024). In a different direction, fold-bifurcation classification is achieved by embedding physics in the data pipeline: polar amplitude extraction, normalization, logarithmic scaling, and moving-mean filtering produce envelope-like signals that expose critical slowing down near a fold of cycles, after which a shallow 1D CNN trained on one oscillator generalizes to three other systems (Habib et al., 2023). A plausible implication is that, within the PInteCNN umbrella, the relevant invariant may reside in the representation presented to the CNN rather than in the final loss.

6. Empirical profile, automation, and limitations

The empirical record is heterogeneous because the category itself is heterogeneous. PIDE-inspired nonlocal CNNs achieve $\partial_y$6 on CIFAR-10, $\partial_y$7 on CIFAR-100, and $\partial_y$8 mIoU on BDD100K, while maintaining moderate parameter counts and using two-stage nonlocal blocks as the recommended design (Bohn et al., 2021). IC-based remodeling of convolutional units reduces ResNet-50 ImageNet top-1 error from $\partial_y$9 to $$(\mathcal{K}u)(x)=\int_{\Omega} K(x,y)\big(u(y)-u(x)\big)\,dy,$$0 in one formulation, and a later IC-network paper reports $$(\mathcal{K}u)(x)=\int_{\Omega} K(x,y)\big(u(y)-u(x)\big)\,dy,$$1 top-1 error with weak logit distillation, matching ResNet-100 with nearly half of FLOPs (An et al., 2019). PhysNet improves medical-image classification with interpretable latent fields and biologically plausible learned parameters (Das et al., 30 Mar 2026). Auto-PICNN shows that automated loss and architecture selection can materially improve PDE surrogates without manual design (Zhou et al., 2024).

Several limitations recur. Dense nonlocal operators incur $$(\mathcal{K}u)(x)=\int_{\Omega} K(x,y)\big(u(y)-u(x)\big)\,dy,$$2 affinity costs unless subsampled or otherwise compressed (Bohn et al., 2021). Irregular-domain PICNNs presently rely on smooth, fold-free mappings with $$(\mathcal{K}u)(x)=\int_{\Omega} K(x,y)\big(u(y)-u(x)\big)\,dy,$$3, and the reported evaluation is qualitative, without $$(\mathcal{K}u)(x)=\int_{\Omega} K(x,y)\big(u(y)-u(x)\big)\,dy,$$4 errors or ablation studies (Zhou et al., 10 Jun 2025). MRF-PINN and related convolutional PINNs remain tied to regular or logically rectangular grids and to the specific boundary formulations handled by their padding strategies (Zhang et al., 2022). Rule-guided detectors depend on the quality of scene segmentation, class balance, and LLM-derived rules, and can improve IoU while not improving mAP on all datasets (Gupta et al., 2024). Physics-inspired CNN modules such as IC layers integrate physical analogy into the operator form but do not enforce conservation laws or PDE residuals (An et al., 2019).

A second misconception is that PInteCNNs are inherently more interpretable than other models. Some are: PhysNet exposes latent fields and learned diffusion and growth rates, and PDE-oriented PICNNs expose residual maps, derivative fields, or source states (Das et al., 30 Mar 2026). Others primarily inherit a physics-inspired inductive bias without providing a direct mechanistic readout. The field is therefore best understood as a spectrum. At one end are strict scientific solvers, where convolutional structure is subordinated to numerical residuals and geometry handling; at the other are CNNs whose physics enters as architectural priors, rules, or transformed observables. What unifies them is not a single architecture, but the decision to make physical structure a first-class component of convolutional computation rather than an external afterthought.