SCNO: Spiking Compositional Neural Operator -- Towards a Neuromorphic Foundation Model for Nuclear PDE Solving

Samrendra Roy; Souvik Chakraborty; Rizwan-uddin; Syed Bahauddin Alam

cs.LGApr 13, 2026

SCNO: Spiking Compositional Neural Operator -- Towards a Neuromorphic Foundation Model for Nuclear PDE Solving

Authors: Samrendra Roy, Souvik Chakraborty, Rizwan-uddin, Syed Bahauddin Alam

Organizations: University of Illinois Urbana-Champaign Department of Nuclear, Plasma & Radiological Engineering Urbana, IL, USA · Indian Institute of Technology Delhi Department of Applied Mechanics New Delhi, India · National Center for Supercomputing Applications Urbana, IL, USA

Abstract

Neural operators have emerged as powerful surrogates for partial differential equation (PDE) solvers, yet they are typically trained as monolithic models for individual PDEs, require energy-intensive GPU hardware, and must be retrained from scratch when new physics emerge. We introduce the Spiking Compositional Neural Operator (SCNO), a modular architecture combining spiking and conventional components that addresses all three limitations. SCNO maintains a library of small spiking neural operator blocks, each trained on a single elementary differential operator (convection, diffusion, reaction), and composes them through a lightweight input-conditioned aggregator to solve coupled PDEs not seen during block training. A small correction network learns cross-coupling residuals while keeping all blocks and the aggregator frozen, preserving zero-forgetting modular expansion by construction. We evaluate SCNO on eight PDE families including five coupled systems and a nuclear-relevant 1-group neutron diffusion equation. SCNO with correction achieves the lowest relative $L^2$ error on four of five coupled PDEs, outperforming both a monolithic spiking DeepONet (by up to 62%, mean over 3 seeds) and a standard ANN DeepONet (by up to 65%), while requiring only 95K trainable parameters versus 462K for the monolithic baseline. To our knowledge, this is the first compositional spiking neural operator and the first proof-of-concept for modular neuromorphic PDE solving with built-in forgetting-free expansion.

Figures & tables

Panel of Figure 1 — Figure 1. SCNO architecture. Frozen spiking operator blocks ( $\mathcal{B}_{\nabla}$ , $\mathcal{B}_{\Delta}$ , $\mathcal{B}_{f}$ ), each containing three LIF layers with skip connections, are composed through an input-conditioned gated aggregator. A correction network ( $\mathcal{C}$ , 95K params) learns cross-coupling residuals scaled by learnable $\alpha$ . Blocks remain frozen, guaranteeing zero forgetting upon modular expansion.

PDE	Equation	Type	Blocks
Convection	$\partial_{t}u+c\partial_{x}u=0$	Elem.	$\nabla$
Diffusion	$\partial_{t}u=\nu\partial_{xx}u$	Elem.	$\Delta$
Reaction	$\partial_{t}u=ku(1{-}u)$	Elem.	$f$
Conv-Diff	$\partial_{t}u+c\partial_{x}u=\nu\partial_{xx}u$	Coupled	$\nabla{+}\Delta$
React-Diff	$\partial_{t}u=\nu\partial_{xx}u+ku(1{-}u)$	Coupled	$f{+}\Delta$
Neutron Diff	$\partial_{t}\phi=D\partial_{xx}\phi+(\nu\Sigma_{f}{-}\Sigma_{a})\phi$	Nuclear	$\Delta{+}f$

Table 1. PDE families and their compositional structure.

Figure 2. Training convergence for elementary spiking blocks. ReduceLROnPlateau with milestone fallback enables smooth convergence over 800 epochs.

Figure 3. Representative predictions from elementary spiking blocks on held-out test samples. Each row corresponds to one block (convection

T_{s}{=}30

, diffusion

T_{s}{=}20

, reaction

T_{s}{=}20

). Gray dashed: initial condition

u_{0}(x)

; blue: ground truth

u(x,T)

; red dashed: SCNO prediction. All blocks achieve sub-10% relative

L^{2}

error on unseen initial conditions.

PDE	SCNO	SCNO+Corr	Mono SNN	ANN
Conv-Diff	28.1 $\pm$ 0.2	14.1 $\pm$ 0.5	10.7 $\pm$ 1.8	15.7 $\pm$ 1.8
React-Diff	2.1 $\pm$ 0.0	2.1 $\pm$ 0.0	5.5 $\pm$ 0.3	6.0 $\pm$ 0.1
Neutron Diff	9.4 $\pm$ 0.0	4.6 $\pm$ 0.1	13.7 $\pm$ 0.9	11.1 $\pm$ 0.8
Burgers	19.6 $\pm$ 0.1	11.6 $\pm$ 0.3	15.3 $\pm$ 1.6	16.2 $\pm$ 1.0
Adv-React	16.7 $\pm$ 0.1	4.1 $\pm$ 0.3	6.2 $\pm$ 0.6	6.2 $\pm$ 0.9
# Train. params	231K	95K	462K	396K

Table 2. Relative

L^{2}

error (%) on coupled PDEs. Bold : best per row. SCNO (frozen) uses only blocks + aggregator. SCNO+Corr adds the correction network. Results averaged over 3 random seeds (

\pm

std). Trainable params per new PDE : SCNO = aggregator (231K); SCNO+Corr = correction (95K). Blocks (462K each) pretrained once, amortized. Mono SNN/ANN = full retrain per PDE.

Figure 4. Nuclear case study: 1-group neutron diffusion

\partial_{t}\phi=D\partial_{xx}\phi+(\nu\Sigma_{f}-\Sigma_{a})\phi

. Top: predictions vs. ground truth. Bottom: pointwise absolute error. SCNO (frozen, shown) achieves 9.4% relative

L^{2}

error; with correction, 4.6%.

Figure 5. Coupled PDE predictions across all five systems and four methods. Each row is a coupled PDE; columns show different test samples. SCNO+Corr (orange dashed) closely tracks the ground truth (blue) on most systems, with the largest residuals appearing on convection-diffusion, the most tightly coupled system.

Figure 6. Comparison across five coupled PDEs: (left) relative

L^{2}

error, (center) parameter counts on log scale, (right) spike counts per sample. “CompNO (frozen)” in the figure legend refers to SCNO without correction. SCNO+Corr achieves the best accuracy on 4/5 PDEs with 5

\times

fewer trainable parameters than the monolithic SNN.

Figure 7. Forgetting-free modular expansion: blocks are added sequentially, and all existing blocks are tested at each phase. Errors are identical across phases—zero forgetting by construction.

Figure 8. Accuracy–efficiency Pareto analysis. SCNO+Corr (orange squares) achieves competitive accuracy with 5

\times

fewer trainable parameters than both the monolithic SNN (red) and ANN (blue) baselines.

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Operator learning is a rapidly advancing area of computational science. It is particularly well suited to problems where a partial differential equation (PDE) must be solved repeatedly under varying physical configurations. Most existing architectures represent the solution operator in a fixed basis. While this assumption is well aligned with global structures, it is less suitable for phenomena governed by local interactions in physical space. We explore an alternative perspective motivated by the observation that the continuum limit of coupled oscillator systems can describe a broad class of PDEs. Building on this idea, we introduce the Kuramoto Neural Operator (KNO), which represents the solution through the evolution of a latent field of interacting oscillators. Across a diverse collection of PDE benchmarks, KNO achieves strong predictive performance, with improvements over competing approaches. Our experimental evaluation also includes an extensive ablation study that quantifies the contribution of each architectural component incorporated into KNO. Furthermore, we show that the model's prediction error is closely linked to the collective dynamics of the latent oscillators. It varies systematically with their degree of synchronization, providing insights into the underlying mechanisms.

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