Compositional Neural Operators (CompNO)

• Compositional Neural Operators (CompNO) are a modular framework that decomposes the evolution of PDEs into reusable neural blocks corresponding to fundamental differential operators.
• The method pretrains separate Fourier Neural Operators for elementary tasks like convection, diffusion, and nonlinear convection, then assembles them for coupled PDE solutions.
• Empirical results show that CompNO achieves superior data efficiency, reduced parameter counts, and enhanced physical interpretability compared to monolithic architectures.

Compositional Neural Operators (CompNO) are a modular framework for scientific foundation modeling of parametric partial differential equations (PDEs), designed to overcome the computational and interpretability limitations of monolithic neural operator architectures. By decomposing the learning of evolution operators for complex PDE systems into reusable neural blocks that correspond to fundamental differential operators, CompNO enables efficient pretraining, flexible assembly, and rigorous enforcement of boundary conditions. This approach has demonstrable advantages in data efficiency, physical interpretability, and adaptability across diverse parameter regimes.

1. Framework Overview and Architectural Components

CompNO is formulated under a “pretrain–assemble–fine-tune” paradigm for foundation modeling of parametric PDEs (Hmida et al., 12 Jan 2026). The architecture consists of three key components: Foundation Blocks, Adaptation Blocks, and Boundary-Condition Operators.

Foundation Blocks

Each Foundation Block is a parametric Fourier Neural Operator (FNO), trained to approximate the temporal evolution $u(t) \mapsto u(t+\Delta t)$ for a specific elementary differential operator—such as convection, diffusion, or nonlinear convection. The FNO paradigm maps functions $u(\cdot)$ via truncated Fourier transforms, channel-wise linear mappings $W$, nonlinearities $\sigma$, and inverse Fourier transforms: $$G(u)(x) = \mathcal F^{-1}\Bigl(\sigma\bigl(\mathcal F(W u + b)\bigr)\Bigr)(x)$$ To handle PDE parameters (e.g., velocity $\beta$, viscosity $\nu$), CompNO appends these parameters as extra channels, constructing parametric input features $f^*(x)$ and propagating these through $L$ stacked Fourier layers, each with tensorized spectral mixing and skip connections.

Specialized Foundation Blocks

• Convection Block: Trained on $\partial_t u + \beta \cdot \nabla u = 0$ with $\beta$ sampled from $\{0.1, 0.4, 0.7, 1, 2\}$.
• Diffusion Block: Trained on $\partial_t u - \nu \Delta u = 0$, $\nu \in \{0.01,0.1,0.2,0.5,1,2\}$.
• Nonlinear Convection Block: Trained on $\partial_t u + u \cdot \nabla u = 0$.

Each block is pretrained independently and its parameters $\theta_i$ are frozen after pretraining.

Adaptation Blocks (Aggregator)

Adaptation Blocks are shallow task-specific neural networks that assemble one or more Foundation Blocks into a solver for a coupled PDE: $$\mathcal S(u; \mathbf p) = A_m \circ O_m (\cdots A_2 \circ O_2 (A_1 \circ O_1 (u; p_1); p_2) \cdots; p_m)$$ For linear systems (e.g., convection-diffusion), $A$ may be a linear aggregator: $$w(x) = \alpha_1 u_{\mathrm{conv}}(x) + \alpha_2 u_{\mathrm{diff}}(x)$$ with $\alpha_1, \alpha_2$ learned to minimize the PDE residual. For nonlinear systems (e.g., Burgers’ equation), a nonlinear MLP aggregator is necessary to represent cross-terms that cannot be captured by linear combinations.

Boundary-Condition Operator

A dedicated Boundary-Condition Operator enforces Dirichlet boundary values exactly at inference time, employing the BOON procedure. This involves boundary response computation of the FNO kernel, pre-correction of values at endpoints, and explicit overwriting to guarantee zero discrepancy with prescribed boundary data. Unlike baseline neural operators (e.g., PFNO, PDEFormer), this ensures exact satisfaction of boundary constraints in all experiments.

2. Training Regime, Losses, and Evaluation Metrics

The CompNO pipeline separates pretraining of Foundation Blocks from fine-tuning of Adaptation Blocks:

• Foundation Block Pretraining: Each block is trained for 1000 epochs (batch size 50) with Adam (initial learning rate $10^{-3}$, StepLR $\gamma=0.5$ every 100 epochs) to minimize mean squared error (MSE) loss:

$$\text{MSE} = \frac{1}{T} \sum_{t=1}^T \|u_t - \hat u_t\|^2$$

Training data includes 8,000 instances per operator; 2,000 are reserved for testing.

• Adaptation Block Fine-tuning: With Foundation Blocks' weights frozen, only aggregator parameters are optimized for 100 epochs (batch size 4, Adam, learning rate $10^{-4}$). Mean absolute error (MAE) is used:

$$\text{MAE} = \frac{1}{T} \sum_{t=1}^T \|u_t - \hat u_t\|_1$$

MAE is selected over MSE to stabilize gradients in few-shot adaptation regimes.

• Evaluation Metric: Relative $L^2$ error is used to assess prediction accuracy:

$$\text{RelErr} = \frac{\|u_{\rm pred} - u_{\rm true}\|_2}{\|u_{\rm true}\|_2}$$

or, time-averaged with normalization.

3. Numerical Results and Empirical Comparison

CompNO was evaluated on the PDEBench one-dimensional suite, including convection, diffusion, convection–diffusion, and Burgers’ equations. The comparison metrics include test MAE, MSE, and relative $L^2$ errors; baseline models include PFNO, PDEFormer, and in-context learning (ICL) based models.

Pretraining Quality

Foundation Block	Parameters	MSE train ($\times 10^{-4}$)	MSE test ($\times 10^{-4}$)
Convection PFNO	217k	0.95	0.98
Diffusion PFNO	217k	5.7	5.8
Nonlinear-Convection FNO	75k	7.6	8.0

Aggregator Adaptation (Coupled PDEs)

Problem	CompNO Params	CompNO Train MAE	CompNO Test MAE ($N_x=2048$)	Boundary Error	PFNO Params	PFNO Train MAE	PFNO Boundary Error
Convection–Diffusion	257	0.008	0.019	0	217k	0.039	1.53
Burgers’	132k	0.01	0.03	0	217k	0.002	0.01

Relative $L^2$ Test Error (PDEBench)

PDE	PDEFormer-FS	ICL-based	CompNO
Convection	0.0124	0.0184	0.0071
Diffusion	—	0.0120	0.0075
Convection–Diffusion	—	0.0231	0.0120
Burgers ($\nu=0.1$)	0.0135	—	0.0280
Burgers ($\nu=0.01$)	0.0399	—	0.0371

4. Generalization, Boundary Satisfaction, and Interpretability

CompNO demonstrates robust generalization to broader parameter regimes. For convection–diffusion, it achieves the best accuracy in diffusion-dominated, low-Peclet-number flows (Pe $\leq$ 2). Error increases in advection-dominated regimes but remains competitive. For Burgers’ equation, test error is stable for low Reynolds numbers (Re $\leq$ 1), peaks in the transition regime (Re $\approx$ 10–100), and partially recovers at high Re, indicating an ability to resolve large-scale structures but only approximate rapid inertial features.

Boundary satisfaction is exact for Dirichlet conditions; boundary error is zero by construction in all experiments, which further reduces interior prediction error (MAE decreases from 0.025 to 0.004 when BCs are enforced). This distinguishes CompNO fundamentally from baselines, for which endpoint mismatches occur. The model trained on a $1024$-point grid generalizes to $2048$ points with only moderate MAE increase, confirming operator- rather than grid-based learning.

Interpretability is enhanced because each Foundation Block corresponds to a known mathematical operator (such as $\nabla$, $\Delta$, or a nonlinear term), and the aggregation weights directly reflect a physics-informed decomposition of the target evolution operator.

5. Data Efficiency, Scalability, and Limitations

CompNO is highly data-efficient: Foundation Blocks only require pretraining on elementary operators, which is computationally cheap. Adaptation Blocks are amenable to few-shot fine-tuning since they comprise only a handful of parameters for linear aggregation or small MLPs for nonlinear aggregation. This yields substantial parameter count reduction compared to monolithic networks: e.g., CompNO-v1 for convection–diffusion with 257 parameters versus PFNO's 217k.

The architecture is mesh-invariant and dimension-agnostic, due to its reliance on spectral FNOs, suggesting direct extensions to 2D and 3D PDEs. Nonetheless, such extensions may require more expressive aggregation schemes and new boundary-enforcement strategies.

All reported experiments are one-dimensional. Current adaptation blocks may underfit in scenarios with strong nonlinearity or developed turbulence, especially as Pe or Re increases. A plausible implication is that performance on turbulent or high-dimensional flows may be bottlenecked by the expressivity of both the operator library and the chosen aggregator.

6. Future Directions and Research Outlook

Planned improvements for CompNO include enriching the Foundation Block library to encompass additional operators (e.g., pressure-gradient, divergence), exploring hierarchical or variational aggregator architectures, and systematically validating the pipeline on multi-dimensional Navier–Stokes systems and complex geometries. Such developments are intended to establish CompNO as a general-purpose PDE foundation model, scalable across domains and adaptable to both linear and strongly nonlinear systems (Hmida et al., 12 Jan 2026).