Physics-Informed Convolutional Neural Operator

• PICNO is a physics-informed neural operator that replaces low-rank kernels with convolutional layers to learn PDE solutions, demonstrated in Helmholtz wavefield settings.
• Its hybrid loss framework combines supervised data matching with PDE residual penalties to achieve improved high-frequency wavefield prediction accuracy.
• PICNO’s design ensures resolution-invariance on regular grids, offering rapid predictions that outperform traditional numerical solvers in complex acoustic simulations.

Physics-Informed Convolutional Neural Operator (PICNO) denotes a physics-informed neural operator whose internal operator layers are implemented as convolutions, for example in the style of CNO or FNO, rather than general low-rank point-cloud kernels. In the most explicit formulation currently described under this name, PICNO is introduced for the Helmholtz equation: it takes the background wavefield corresponding to a homogeneous medium and the velocity model as input function space, and generates the scattered wavefield as the output function space (Ma et al., 22 Jul 2025). The term is not fully standardized across the literature: in the low-rank neural-operator literature, PICNO is described as the convolutional counterpart of a physics-informed low-rank neural operator, preserving the operator-learning viewpoint and the physics-informed penalty framework while replacing low-rank kernel integration by convolution (Schaffer et al., 9 Sep 2025).

1. Definition and conceptual boundaries

PICNO belongs to the broader class of neural operators, whose aim is to approximate a solution operator

$\mathcal{T}: \mathcal{F}\to\mathcal{U},$

mapping an input function such as a source term, coefficient field, parameter field, or initial condition to the corresponding PDE solution. In the convolutional specialization, the operator layer is written in translation-invariant form,

$$v_{t+1}(x)=\sigma\big(W_t v_t(x)+(\kappa_t * v_t)(x)\big),$$

so the nonlocal interaction is implemented by convolution rather than by a general kernel $k_t(x,y)$ (Schaffer et al., 9 Sep 2025).

A useful boundary distinction is between PICNO and physics-informed convolutional neural networks (PICNNs). PICNNs replace the MLP of a PINN by a CNN and focus on parametric PDEs on grids, with derivatives computed numerically by fixed convolution kernels. Auto-PICNN states that the missing ingredient to call such models a full neural operator is primarily that the architecture is still image-CNN based and not explicitly designed to be resolution-invariant or mesh-independent (Zhou et al., 2024). By contrast, PICNO is explicitly framed as an operator learner.

This distinction also separates PICNO from single-instance physics-informed convolutional solvers. The earlier “physics-informed convolutional network” (PICN) generates a field with a shallow deconvolution-plus-convolution architecture and enforces PDE residuals and boundary conditions through fixed differential filters and interpolation. That framework is already close in spirit to PICNO, but it is primarily organized as a solver for fixed PDE instances rather than as a general operator learner over families of inputs (Shi et al., 2022).

2. Operator formulations and architectural patterns

The clearest operator-level PICNO instantiation appears in the Helmholtz-wavefield setting. There, the learned map is

$$\mathcal{G}:(U_0^{\mathrm{Re}},U_0^{\mathrm{Im}},v)\mapsto (\delta U^{\mathrm{Re}},\delta U^{\mathrm{Im}}),$$

where $U_0$ is the background wavefield in a homogeneous medium, $v(x,z)$ is the acoustic velocity, and $\delta U$ is the scattered wavefield (Ma et al., 22 Jul 2025). The backbone is a CNO with a U-Net-style multiscale organization built from downsampling blocks, upsampling blocks, invariant blocks, and residual blocks. The input is a three-channel 2-D field consisting of $U_0^{\mathrm{Re}}, U_0^{\mathrm{Im}}, v$, and the output is a two-channel field $\delta U^{\mathrm{Re}}, \delta U^{\mathrm{Im}}$ (Ma et al., 22 Jul 2025).

The PILNO formulation provides a generic architectural translation rule for PICNO. PILNO learns

$\mathcal{T}: \mathcal{F}\to\mathcal{U}$

through low-rank kernels $$v_{t+1}(x)=\sigma\big(W_t v_t(x)+(\kappa_t * v_t)(x)\big),$$0, an encoder–decoder organization on point clouds, and a physics-informed penalty loss. The paper explicitly states that a PICNO would keep the operator-learning view, the encoder–decoder design, and the physics-informed penalty framework, but would replace low-rank point-cloud kernels by translation-invariant convolutional kernels, evaluated by FFT or standard convolutions on regular grids (Schaffer et al., 9 Sep 2025). In that sense, PICNO is not a single fixed architecture but a design pattern: convolutional operator blocks plus physics-informed training.

Conditioning mechanisms are also inherited directly. In the screened Poisson example of the low-rank formulation, the operator is conditional on a scalar parameter $$v_{t+1}(x)=\sigma\big(W_t v_t(x)+(\kappa_t * v_t)(x)\big),$$1; in the Darcy example it is conditional on a field $$v_{t+1}(x)=\sigma\big(W_t v_t(x)+(\kappa_t * v_t)(x)\big),$$2. The same design is explicitly described as transferable to PICNO, with scalar parameters entering as extra channels or global conditioning variables and coefficient fields entering as additional input channels (Schaffer et al., 9 Sep 2025).

3. Physics-informed loss constructions

The defining feature of PICNO is that operator learning is constrained by the governing PDE rather than by supervised field matching alone. In the Helmholtz formulation, the scattered-field equation is

$$v_{t+1}(x)=\sigma\big(W_t v_t(x)+(\kappa_t * v_t)(x)\big),$$3

and the physics-informed loss is

$$v_{t+1}(x)=\sigma\big(W_t v_t(x)+(\kappa_t * v_t)(x)\big),$$4

This is combined with a supervised data term

$$v_{t+1}(x)=\sigma\big(W_t v_t(x)+(\kappa_t * v_t)(x)\big),$$5

into

$$v_{t+1}(x)=\sigma\big(W_t v_t(x)+(\kappa_t * v_t)(x)\big),$$6

with $$v_{t+1}(x)=\sigma\big(W_t v_t(x)+(\kappa_t * v_t)(x)\big),$$7 tuned empirically (Ma et al., 22 Jul 2025).

The general penalty form is broader. In the parametric elliptic operator setting, the PDE is written as

$$v_{t+1}(x)=\sigma\big(W_t v_t(x)+(\kappa_t * v_t)(x)\big),$$8

and the operator prediction $$v_{t+1}(x)=\sigma\big(W_t v_t(x)+(\kappa_t * v_t)(x)\big),$$9 is trained with a PDE residual term $k_t(x,y)$0, a boundary penalty $k_t(x,y)$1, and a combined physics-informed loss

$k_t(x,y)$2

The same formulation is stated to transfer directly to PICNO once the low-rank operator is replaced by convolutional layers (Schaffer et al., 9 Sep 2025).

More elaborate loss constructions already exist in adjacent physics-informed operator models. PI-HAMNO uses a multi-objective strategy combining data fitting with strong-form and weak-form physics constraints. Its total loss is

$k_t(x,y)$3

with strong-form residuals evaluated in physical coordinates and weak-form residuals assembled using finite-element test functions and centroid-based tetrahedral quadrature (Bamdad et al., 10 Jun 2026). This suggests that PICNO need not be limited to a single residual penalty; strong-form, weak-form, and hybrid formulations are all compatible with convolutional operator backbones.

Derivative evaluation varies by formulation. Helmholtz PICNO uses an 8th-order finite-difference stencil for the Laplacian rather than automatic differentiation (Ma et al., 22 Jul 2025). Auto-PICNN similarly computes spatial derivatives numerically using non-trainable convolution kernels such as Sobel filters and central-difference stencils (Zhou et al., 2024). By contrast, the low-rank point-cloud formulation emphasizes that encoder–decoder structure simplifies automatic differentiation because derivatives are only needed through the decoder, not through the whole encoding pipeline (Schaffer et al., 9 Sep 2025).

4. Geometry, temporal evolution, and multiscale variants

PICNO is most natural on regular grids, where translation invariance makes convolutional kernels well defined. This is precisely the regime contrasted with point-cloud operator learning: PILNO emphasizes that point-cloud methods are discretization-independent on arbitrary sensor sets, whereas a convolutional neural operator assumes structured grids and implements the integral operator via FFT or discrete convolutions (Schaffer et al., 9 Sep 2025). A plausible implication is that PICNO is strongest when the PDE domain and sampling admit a regular-grid representation, while related operator families become preferable on irregular geometries.

Temporal and multiscale generalizations already appear in neighboring operator architectures. HAMNO combines local convolutional representations, global spectral operators, and a hierarchical encoder–decoder; its physics-informed extension, PI-HAMNO, adds strong-form and weak-form losses for time-dependent PDEs on cubic domains. The HAMNO analysis explicitly states that if the global spectral branch is turned off, the remaining model is a deep hierarchical U-Net–like convolutional neural operator, which is essentially what one might call a PICNO when equipped with the same physics-informed losses (Bamdad et al., 10 Jun 2026). In that formulation, the operator learns one-step maps and is deployed autoregressively.

For partitioned coupled systems, the “Physics-Informed Partitioned Coupled Neural Operator” (PCNO) provides a related but more specialized construction. PCNO designs a joint convolution operator within the Fourier layer to capture all sub-regions simultaneously and introduces grid alignment layers so that the coupling relationship between sub-regions can be learned in the frequency domain (Wu et al., 2024). Although PCNO is not presented as a generic PICNO, it is explicitly described as a strong candidate for a “Physics-Informed Convolutional Neural Operator” for partitioned, coupled PDE networks.

For irregular and variable geometries, the literature also points beyond pure grid convolution. The physics- and geometry-aware spatio-spectral graph neural operator $k_t(x,y)$4G-Sp$k_t(x,y)$5GNO uses graph spectral and spatial operators, geometry encoders, and a hybrid time-marching physics-informed loss to learn operators on complex domains and under geometry variation at inference (Sarkar et al., 13 Aug 2025). This suggests that the PICNO idea has close relatives in graph-based operator learning whenever the geometry no longer supports a clean convolutional grid.

5. Applications and empirical evidence

The first explicit PICNO application is 2-D, constant-density, isotropic acoustic wave propagation in the frequency domain. The governing Helmholtz equation is rewritten in a scattered-field form,

$k_t(x,y)$6

with $k_t(x,y)$7 given by the 2-D Green’s function in a homogeneous medium,

$k_t(x,y)$8

so source position and frequency are encoded analytically through the background field rather than passed as separate scalars (Ma et al., 22 Jul 2025).

In the reported Case 1 experiments, the frequencies are 8 Hz, 10 Hz, and 12 Hz. The relative $k_t(x,y)$9 errors are

• CNO: 0.25, 0.34, 0.50
• PICNO: 0.18, 0.22, 0.23

with relative promotion of 27.4%, 35.2%, and 53.1%, respectively (Ma et al., 22 Jul 2025). The improvement grows with frequency, which the paper emphasizes as particularly important for high-frequency wavefield prediction. The same study reports that a conventional numerical solver requires about 3 seconds per frequency-domain forward solve, whereas both CNO and PICNO produce a prediction in about 0.83 seconds once trained (Ma et al., 22 Jul 2025).

The second Helmholtz setting places sources throughout the entire domain rather than fixing source depth. In that more challenging configuration, PICNO again attains much lower PDE loss and significantly lower validation error than a purely data-driven CNO at 12 Hz and 14 Hz (Ma et al., 22 Jul 2025). An out-of-distribution test with faults and sharp discontinuities shows that CNO produces large errors and unphysical wave patterns, whereas PICNO remains close to the reference solution, though somewhat smoother (Ma et al., 22 Jul 2025).

Related evidence from adjacent frameworks supports broader PICNO design choices. PI-HAMNO improves stability, physical consistency, and data efficiency over data-only baselines in long-horizon rollout and out-of-distribution tests for Allen–Cahn, Cahn–Hilliard, and Swift–Hohenberg dynamics (Bamdad et al., 10 Jun 2026). Auto-PICNN reports that physics-informed convolutional models can materially outperform manually designed baselines across heat, Poisson, Darcy, and Navier–Stokes benchmarks, underscoring that loss design and architecture search are central even before explicit operator-theoretic generalization is imposed (Zhou et al., 2024).

6. Relations, limitations, and research directions

A common misconception is that any physics-informed CNN for PDEs is automatically a PICNO. The current literature is more precise. PICNNs are parametric CNN surrogates with physics-informed losses, but Auto-PICNN explicitly notes that the missing ingredient to call them full neural operators is that they are not explicitly designed to be resolution-invariant or mesh-independent (Zhou et al., 2024). PICNO, by contrast, is an operator-learning formulation.

Another misconception is that physics-informed operator learning must be fully unsupervised. The explicit Helmholtz PICNO is not: it uses a hybrid loss $$\mathcal{G}:(U_0^{\mathrm{Re}},U_0^{\mathrm{Im}},v)\mapsto (\delta U^{\mathrm{Re}},\delta U^{\mathrm{Im}}),$$0 (Ma et al., 22 Jul 2025). PCNO likewise combines data and PDE loss, and its results show that pure PDE loss yields much larger errors on complex coupled networks (Wu et al., 2024). Fully unsupervised operator learning is possible in other architectures, but it is not definitional for PICNO.

Current limitations are also explicit. The Helmholtz PICNO is restricted to 2-D, constant-density, isotropic acoustic media, uses a constant background velocity $$\mathcal{G}:(U_0^{\mathrm{Re}},U_0^{\mathrm{Im}},v)\mapsto (\delta U^{\mathrm{Re}},\delta U^{\mathrm{Im}}),$$1, trains separately for each frequency, and treats boundary conditions implicitly through the training data rather than as separate architectural inputs (Ma et al., 22 Jul 2025). Training is sensitive to the loss weight $$\mathcal{G}:(U_0^{\mathrm{Re}},U_0^{\mathrm{Im}},v)\mapsto (\delta U^{\mathrm{Re}},\delta U^{\mathrm{Im}}),$$2, and the validation curves can show abrupt fluctuations and occasional jumps because the PDE term induces a stiff optimization landscape (Ma et al., 22 Jul 2025).

Several neighboring lines of work point to likely future directions. Multi-stage curriculum training for physics-informed neural operators progressively enforces boundary conditions and then interior residuals, with optimizer re-initialization acting as a continuation mechanism; this is described as a robust paradigm for physics-informed operator learning (Marcandelli et al., 2 Feb 2026). A plausible implication is that staged optimization is directly relevant for PICNO training, especially when PDE residuals make the loss landscape unstable. Likewise, pseudo-physics-informed neural operators construct a surrogate PDE system from rudimentary differential operators when the true governing equations are unavailable, and use alternating updates to regularize operator learning in data-scarce regimes (Chen et al., 4 Feb 2025). This suggests a route for PICNO-like models in settings where only partial physical knowledge is available.

Taken together, the literature presents PICNO not as a single canonical network but as a family of operator-learning constructions in which convolutional operator layers are trained under PDE constraints. The most explicit current realization is a CNO-based scattered-field Helmholtz solver (Ma et al., 22 Jul 2025); the broader literature supplies the corresponding operator-theoretic abstraction, alternative loss formulations, multiscale and temporal extensions, and geometry-aware generalizations (Schaffer et al., 9 Sep 2025).