Composite Operator-Based Neural Network (CPNN)

• CPNN is defined as an operator-learning paradigm where mappings are represented by coordinated sub-operators rather than a single monolithic network.
• It employs an encoder-decoder architecture to separately process input groups and fuse them for accurate prediction of complex hydrodynamic and particulate phenomena.
• Empirical results show CPNN's superior efficiency and accuracy in modeling stormwater flow and pollutant dispersion using physics-resolved simulation data.

Searching arXiv for the cited CPNN-related papers to ground the article in recent literature. arXiv search: (Shatarah et al., 7 Jul 2025) Composite Operator-Based Neural Network stormwater dynamics Composite Operator-Based Neural Network (CPNN) denotes, in the literature surveyed here, a composite operator-learning paradigm in which a target mapping is represented by coordinated sub-operators rather than by a single monolithic network. In its explicit stormwater-infrastructure formulation, CPNN combines a multi-input operator encoder with a fully connected decoder to predict unsteady three-dimensional hydrodynamics and particulate matter transport in a hydrodynamic separator (Shatarah et al., 7 Jul 2025). Related work uses closely aligned composite-neural-operator constructions for multiscale bubble growth, structural seismic response prediction, and dual-path neural operators, which suggests that CPNN is best understood as an architectural principle centered on operator composition, separable encoding, and learnable fusion rather than as one universally fixed model family (Lu et al., 2024, Kim et al., 12 Jun 2025, Wang et al., 17 Jul 2025).

1. Terminology and conceptual scope

Within the surveyed literature, the most explicit use of the term “Composite Operator-Based Neural Network” appears in a stormwater-treatment study that casts prediction as learning nonlinear operators from parameterized event descriptors, particulate class, time, and spatial coordinates to spatiotemporal output fields (Shatarah et al., 7 Jul 2025). That work uses the acronym CPNN for a concrete architecture, not merely for a loose design metaphor.

A broader antecedent appears in the “composite neural network” framework, where pre-trained and non-instantiated neural modules are arranged as a rooted directed acyclic graph, with linear or affine combiners and pointwise activations connecting heterogeneous components (Yang et al., 2019). In that formulation, pre-trained modules are treated as frozen operators and only a small number of combiner parameters are learned end-to-end. This provides an operator-centric interpretation of composite modeling even though the paper does not use the exact CPNN acronym.

The nomenclature is not uniform across fields. A neuroevolution paper explicitly states that “Composite Operator-Based Neural Network (CPNN) does not correspond to a standard model in the neuroevolution literature” and that, in that context, the phrase is “almost certainly a misnomer for Compositional Pattern Producing Networks (CPPNs)” (Fernando et al., 2016). This distinction is important because operator-based CPNN and coordinate-generating CPPN describe different research lineages.

2. Mathematical formulation as operator composition

In the stormwater formulation, the task is cast as learning a nonlinear operator

$\mathcal{M}:\mathbb{X}\rightarrow \mathbb{Y},$

where $\mathbb{X}$ comprises parameterized inputs and $\mathbb{Y}$ comprises outputs over all cases, times, and spatial samples (Shatarah et al., 7 Jul 2025). Two separate CPNNs are trained: one for hydrodynamics and one for particulate-matter transport. The hydraulic operator is written as

$$O_{\mathrm{hyd}}:\mathcal{X}_{\mathrm{hyd}} \rightarrow |\mathbf{u}|(\mathbf{x},t),$$

with $$\mathcal{X}_{\mathrm{hyd}}=\{\mathbf{p},\mathbf{T},\mathbf{q}\}$$, $\mathbf{p}=[\lambda,k,\theta,C_0,k_d]$, $\mathbf{T}=[t]$, and $\mathbf{q}=[x,y,z]$. The PM operator is written as

$$O_{\mathrm{PM}}:\mathcal{X}_{\mathrm{PM}} \rightarrow c(\mathbf{x},t),$$

with $$\mathcal{X}_{\mathrm{PM}}=\{\mathbf{p},\mathbf{cl},\mathbf{T},\mathbf{q}\}$$ and $\mathbb{X}$0.

The paper positions CPNN against three increasingly structured operator parameterizations: $\mathbb{X}$1

$\mathbb{X}$2

$\mathbb{X}$3

where $\mathbb{X}$4 denotes Cartesian product over inputs and $\mathbb{X}$5 denotes element-wise Hadamard product over latent embeddings. The CPNN extends the MIONet encoder with a decoder: $\mathbb{X}$6

$\mathbb{X}$7

with $\mathbb{X}$8 the MIONet encoder and $\mathbb{X}$9 the activation; Tanh is used, and Leaky ReLU is also investigated.

A related operator-composition formalism appears in dual-path neural operators, where a residual path and a dense path are run in parallel: $\mathbb{Y}$0 followed by

$\mathbb{Y}$1

This parallel composite view is not identical to the stormwater CPNN, but it formalizes the same core idea that complex solution operators can be constructed from simpler operator blocks via composition and fusion (Wang et al., 17 Jul 2025).

3. Architectural realization in the stormwater CPNN

The stormwater CPNN uses an encoder-decoder decomposition in which two branch networks encode event parameters $\mathbb{Y}$2 and PM class $\mathbb{Y}$3, and two trunk networks encode time $\mathbb{Y}$4 and spatial coordinates $\mathbb{Y}$5 (Shatarah et al., 7 Jul 2025). Their latent outputs are merged by a Hadamard product and passed to a fully connected decoder. The architecture is explicitly “composite” because the input groups are embedded separately and only then composed.

The encoder structure is motivated by operator learning rather than by physics-informed residual minimization. The CFD labels come from unsteady RANS hydrodynamics coupled with Euler–Euler PM transport under one-way coupling, but the CPNN itself is trained without PINN terms. The PM operator does not explicitly take predicted $\mathbb{Y}$6 as input; instead, both hydraulic and PM operators share the same encoder inputs, and PM transport dependencies on hydraulics are learned implicitly from the CFD-generated labels.

Input group	Symbol	Mini-batch shape
Event parameters	$\mathbb{Y}$7	$\mathbb{Y}$8
PM class	$\mathbb{Y}$9	$$O_{\mathrm{hyd}}:\mathcal{X}_{\mathrm{hyd}} \rightarrow |\mathbf{u}|(\mathbf{x},t),$$0
Time	$$O_{\mathrm{hyd}}:\mathcal{X}_{\mathrm{hyd}} \rightarrow |\mathbf{u}|(\mathbf{x},t),$$1	$$O_{\mathrm{hyd}}:\mathcal{X}_{\mathrm{hyd}} \rightarrow |\mathbf{u}|(\mathbf{x},t),$$2
Spatial coordinates	$$O_{\mathrm{hyd}}:\mathcal{X}_{\mathrm{hyd}} \rightarrow |\mathbf{u}|(\mathbf{x},t),$$3	$$O_{\mathrm{hyd}}:\mathcal{X}_{\mathrm{hyd}} \rightarrow |\mathbf{u}|(\mathbf{x},t),$$4

After broadcasting and Hadamard composition, the merged latent tensor has shape $$O_{\mathrm{hyd}}:\mathcal{X}_{\mathrm{hyd}} \rightarrow |\mathbf{u}|(\mathbf{x},t),$$5. The best Optuna configuration uses $$O_{\mathrm{hyd}}:\mathcal{X}_{\mathrm{hyd}} \rightarrow |\mathbf{u}|(\mathbf{x},t),$$6 MIONet encoding layers, $$O_{\mathrm{hyd}}:\mathcal{X}_{\mathrm{hyd}} \rightarrow |\mathbf{u}|(\mathbf{x},t),$$7 FCNN layers, $$O_{\mathrm{hyd}}:\mathcal{X}_{\mathrm{hyd}} \rightarrow |\mathbf{u}|(\mathbf{x},t),$$8 hidden neurons, learning rate $$O_{\mathrm{hyd}}:\mathcal{X}_{\mathrm{hyd}} \rightarrow |\mathbf{u}|(\mathbf{x},t),$$9, decay rate $$\mathcal{X}_{\mathrm{hyd}}=\{\mathbf{p},\mathbf{T},\mathbf{q}\}$$0, and mini-batch sizes $$\mathcal{X}_{\mathrm{hyd}}=\{\mathbf{p},\mathbf{T},\mathbf{q}\}$$1, $$\mathcal{X}_{\mathrm{hyd}}=\{\mathbf{p},\mathbf{T},\mathbf{q}\}$$2, and $$\mathcal{X}_{\mathrm{hyd}}=\{\mathbf{p},\mathbf{T},\mathbf{q}\}$$3. Two separate networks are used for $$\mathcal{X}_{\mathrm{hyd}}=\{\mathbf{p},\mathbf{T},\mathbf{q}\}$$4 and $$\mathcal{X}_{\mathrm{hyd}}=\{\mathbf{p},\mathbf{T},\mathbf{q}\}$$5, which yielded better PM performance and a smaller memory footprint than a joint network.

4. Governing physics, data generation, and optimization

The stormwater dataset is generated from $$\mathcal{X}_{\mathrm{hyd}}=\{\mathbf{p},\mathbf{T},\mathbf{q}\}$$6 unsteady $$\mathcal{X}_{\mathrm{hyd}}=\{\mathbf{p},\mathbf{T},\mathbf{q}\}$$7D URANS simulations of a full-scale hydrodynamic separator with diameter and depth $$\mathcal{X}_{\mathrm{hyd}}=\{\mathbf{p},\mathbf{T},\mathbf{q}\}$$8, inlet and outlet on the centerline, $$\mathcal{X}_{\mathrm{hyd}}=\{\mathbf{p},\mathbf{T},\mathbf{q}\}$$9 time instances sampled every $\mathbf{p}=[\lambda,k,\theta,C_0,k_d]$0 over one hour, and $\mathbf{p}=[\lambda,k,\theta,C_0,k_d]$1 Latin-hypercube spatial samples per case (Shatarah et al., 7 Jul 2025). Hydrograph and pollutograph parameters are sampled by Latin hypercube sampling over the ranges $\mathbf{p}=[\lambda,k,\theta,C_0,k_d]$2, $\mathbf{p}=[\lambda,k,\theta,C_0,k_d]$3, $\mathbf{p}=[\lambda,k,\theta,C_0,k_d]$4, $\mathbf{p}=[\lambda,k,\theta,C_0,k_d]$5, and $\mathbf{p}=[\lambda,k,\theta,C_0,k_d]$6. The PM phase is represented by nine classes with terminal velocities $\mathbf{p}=[\lambda,k,\theta,C_0,k_d]$7 from $\mathbf{p}=[\lambda,k,\theta,C_0,k_d]$8 to $\mathbf{p}=[\lambda,k,\theta,C_0,k_d]$9.

The paper states that the hydraulics are modeled by incompressible URANS and PM by advection–diffusion with settling and source/sink terms. Canonical forms are given as

$\mathbf{T}=[t]$0

$\mathbf{T}=[t]$1

and

$\mathbf{T}=[t]$2

The learned model is therefore supervised by physics-resolved data, but it is not trained through explicit PDE residual constraints.

The dataset is split into $\mathbf{T}=[t]$3 training cases, $\mathbf{T}=[t]$4 validation cases, and $\mathbf{T}=[t]$5 test cases. Inputs $\mathbf{T}=[t]$6, $\mathbf{T}=[t]$7, $\mathbf{T}=[t]$8, and $\mathbf{T}=[t]$9 are standardized to zero mean and unit variance. Training uses PyTorch with mixed precision, Adam, and standardized MSE. The representative mini-batch loss is

$\mathbf{q}=[x,y,z]$0

Hyperparameter search uses Optuna’s TPE with $\mathbf{q}=[x,y,z]$1 trials, each trained for $\mathbf{q}=[x,y,z]$2 iterations. Baseline comparison models with approximately $\mathbf{q}=[x,y,z]$3 parameters are trained for $\mathbf{q}=[x,y,z]$4 iterations.

A distinctive feature of the framework is its direct use of automatic differentiation. The CPNN can compute

$\mathbf{q}=[x,y,z]$5

as well as derivatives with respect to $\mathbf{q}=[x,y,z]$6, $\mathbf{q}=[x,y,z]$7, and $\mathbf{q}=[x,y,z]$8. In the reported sensitivity analysis, doubling $\mathbf{q}=[x,y,z]$9 accelerates and strengthens the first flush; increasing $$O_{\mathrm{PM}}:\mathcal{X}_{\mathrm{PM}} \rightarrow c(\mathbf{x},t),$$0 and $$O_{\mathrm{PM}}:\mathcal{X}_{\mathrm{PM}} \rightarrow c(\mathbf{x},t),$$1 lowers and delays the hydrograph peak; doubling $$O_{\mathrm{PM}}:\mathcal{X}_{\mathrm{PM}} \rightarrow c(\mathbf{x},t),$$2 uniformly raises concentrations where flow exists; and increasing $$O_{\mathrm{PM}}:\mathcal{X}_{\mathrm{PM}} \rightarrow c(\mathbf{x},t),$$3 has a smaller overall impact.

5. Empirical performance, error structure, and workflow implications

Across dataset splits, the stormwater CPNN reports $$O_{\mathrm{PM}}:\mathcal{X}_{\mathrm{PM}} \rightarrow c(\mathbf{x},t),$$4 in $$O_{\mathrm{PM}}:\mathcal{X}_{\mathrm{PM}} \rightarrow c(\mathbf{x},t),$$5 for concentration $$O_{\mathrm{PM}}:\mathcal{X}_{\mathrm{PM}} \rightarrow c(\mathbf{x},t),$$6 and in $$O_{\mathrm{PM}}:\mathcal{X}_{\mathrm{PM}} \rightarrow c(\mathbf{x},t),$$7 for velocity magnitude $$O_{\mathrm{PM}}:\mathcal{X}_{\mathrm{PM}} \rightarrow c(\mathbf{x},t),$$8 (Shatarah et al., 7 Jul 2025). Case-wise test performance further shows that $$O_{\mathrm{PM}}:\mathcal{X}_{\mathrm{PM}} \rightarrow c(\mathbf{x},t),$$9 of hydraulic test cases have $$\mathcal{X}_{\mathrm{PM}}=\{\mathbf{p},\mathbf{cl},\mathbf{T},\mathbf{q}\}$$0, $$\mathcal{X}_{\mathrm{PM}}=\{\mathbf{p},\mathbf{cl},\mathbf{T},\mathbf{q}\}$$1 fall in the medium category, and none are in the low category; for PM concentration, $$\mathcal{X}_{\mathrm{PM}}=\{\mathbf{p},\mathbf{cl},\mathbf{T},\mathbf{q}\}$$2 of cases have $$\mathcal{X}_{\mathrm{PM}}=\{\mathbf{p},\mathbf{cl},\mathbf{T},\mathbf{q}\}$$3, $$\mathcal{X}_{\mathrm{PM}}=\{\mathbf{p},\mathbf{cl},\mathbf{T},\mathbf{q}\}$$4 lie in $$\mathcal{X}_{\mathrm{PM}}=\{\mathbf{p},\mathbf{cl},\mathbf{T},\mathbf{q}\}$$5, and $$\mathcal{X}_{\mathrm{PM}}=\{\mathbf{p},\mathbf{cl},\mathbf{T},\mathbf{q}\}$$6 are below $$\mathcal{X}_{\mathrm{PM}}=\{\mathbf{p},\mathbf{cl},\mathbf{T},\mathbf{q}\}$$7. The paper uses

$$\mathcal{X}_{\mathrm{PM}}=\{\mathbf{p},\mathbf{cl},\mathbf{T},\mathbf{q}\}$$8

The main baseline comparison is between ANN, MIONet, and CPNN. Eliminating explicit Cartesian-product redundancy reduces the total input footprint from $$\mathcal{X}_{\mathrm{PM}}=\{\mathbf{p},\mathbf{cl},\mathbf{T},\mathbf{q}\}$$9 in ANN to $\mathbb{X}$00 in MIONet and CPNN, a reduction of more than $\mathbb{X}$01. MIONet trains faster, but CPNN is more expressive because the FCNN decoder processes the expanded latent tensor. The reported validation MSE is approximately $\mathbb{X}$02 for CPNN versus approximately $\mathbb{X}$03 for MIONet. ANN and CPNN have similar peak per-batch VRAM, exceeding $\mathbb{X}$04.

The error analysis is explicit about failure modes. Dominant jet flows are learned well, but fine circulation structures in the tank interior can be missed. PM dispersion near inlet-jet impingement on the outlet wall is challenging under complex unsteady loading. Extreme low-flow or very small-$\mathbb{X}$05 events can be mispredicted, including premature settling in the inlet pipe when the network underpredicts $\mathbb{X}$06. The paper attributes this to multiple-orders-of-magnitude variation and to the lower contribution of low-range samples to the total MSE; no explicit loss reweighting is used.

The workflow implications are practical rather than purely methodological. The paper outlines continuous, event-by-event evaluation: long-term hydrograph and pollutograph records are segmented into events, each event is parameterized, CPNN is run per event, and outlet suspended-solids discharge is aggregated into continuous effluent metrics. This establishes CPNN as a surrogate for continuous, long-term performance assessment of stormwater infrastructure rather than only as a one-off emulator of individual CFD simulations.

6. Related formulations, extensions, and conceptual boundaries

Several papers instantiate closely related composite-operator ideas in other scientific domains. In multiscale bubble dynamics, a “composite neural operator model” combines a DeepONet-style mean operator $\mathbb{X}$07 with an LSTM fluctuation generator $\mathbb{X}$08, producing

$\mathbb{X}$09

and reports “99% accuracy for the time evaluation of the bubble radius” while reproducing size-dependent correlated fluctuations (Lu et al., 2024). In structural dynamics, a composite learning framework combines a preprocessing physics operator, a Fourier Neural Operator for discrepancy learning, and a linear-regression postprocessor,

$\mathbb{X}$10

with reported improvements such as top-story RMSE $\mathbb{X}$11 and relative $\mathbb{X}$12 $\mathbb{X}$13 for the five-story Bouc–Wen benchmark (Kim et al., 12 Jun 2025). In PDE operator learning, DPNO composes residual and dense operator paths in parallel and reports relative $\mathbb{X}$14 improvements exceeding $\mathbb{X}$15 on certain Burgers and Darcy benchmarks (Wang et al., 17 Jul 2025).

Work	Composite structure	Reported emphasis
(Shatarah et al., 7 Jul 2025)	MIONet encoder + FCNN decoder	Unsteady hydrodynamics and PM transport
(Lu et al., 2024)	DeepONet mean operator + LSTM fluctuation generator	Multiscale bubble growth with correlated fluctuations
(Kim et al., 12 Jun 2025)	Physics preprocessing + FNO correction + regression postprocessing	Trajectory-level seismic response
(Wang et al., 17 Jul 2025)	Parallel residual and dense operator paths with fusion	PDE operator approximation

A distinct theoretical line defines a composite neural network as a rooted DAG of pre-trained and trainable components, proves that such a network performs better than any individual pre-trained component with a high probability bound, and shows that adding an extra pre-trained component will not degrade overall performance with high probability (Yang et al., 2019). This suggests that composite-operator architectures can be viewed simultaneously as empirical design patterns and as objects of formal study.

A recurring misconception is acronymic rather than architectural. CPNN in the operator-learning sense should not be conflated with CPPN in neuroevolution. The latter refers to Compositional Pattern Producing Networks, a coordinate-based generative encoding later extended to Differentiable Pattern Producing Networks, and the neuroevolution paper explicitly rejects “Composite Operator-Based Neural Network” as a standard label in that literature (Fernando et al., 2016). The distinction is substantive: operator-based CPNN composes scientific surrogate operators, whereas CPPN generates structured patterns or network weights from spatial coordinates.

Taken together, these works indicate that CPNN presently names a family resemblance more than a fully standardized taxonomy. The common invariant is the decomposition of a difficult map into coordinated operators—branches, trunks, physics preprocessors, stochastic fluctuation modules, residual paths, dense paths, or frozen expert modules—followed by learnable recombination.