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
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 L2 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
Figure 1. SCNO architecture. Frozen spiking operator blocks ( B∇ , BΔ , Bf ), each containing three LIF layers with skip connections, are composed through an input-conditioned gated aggregator. A correction network ( C , 95K params) learns cross-coupling residuals scaled by learnable α . Blocks remain frozen, guaranteeing zero forgetting upon modular expansion.
PDE
Equation
Type
Blocks
Convection
∂tu+c∂xu=0
Elem.
∇
Diffusion
∂tu=ν∂xxu
Elem.
Δ
Reaction
∂tu=ku(1−u)
Elem.
f
Conv-Diff
∂tu+c∂xu=ν∂xxu
Coupled
∇+Δ
React-Diff
∂tu=ν∂xxu+ku(1−u)
Coupled
f+Δ
Neutron Diff
∂tϕ=D∂xxϕ+(νΣf−Σa)ϕ
Nuclear
Δ+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 Ts=30 , diffusion Ts=20 , reaction Ts=20 ). Gray dashed: initial condition u0(x) ; blue: ground truth u(x,T) ; red dashed: SCNO prediction. All blocks achieve sub-10% relative L2 error on unseen initial conditions.
PDE
SCNO
SCNO+Corr
Mono SNN
ANN
Conv-Diff
28.1 ± 0.2
14.1 ± 0.5
10.7 ± 1.8
15.7 ± 1.8
React-Diff
2.1 ± 0.0
2.1 ± 0.0
5.5 ± 0.3
6.0 ± 0.1
Neutron Diff
9.4 ± 0.0
4.6 ± 0.1
13.7 ± 0.9
11.1 ± 0.8
Burgers
19.6 ± 0.1
11.6 ± 0.3
15.3 ± 1.6
16.2 ± 1.0
Adv-React
16.7 ± 0.1
4.1 ± 0.3
6.2 ± 0.6
6.2 ± 0.9
# Train. params
231K
95K
462K
396K
Table 2. Relative L2 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 ( ± 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 ∂tϕ=D∂xxϕ+(νΣf−Σa)ϕ . Top: predictions vs. ground truth. Bottom: pointwise absolute error. SCNO (frozen, shown) achieves 9.4% relative L2 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 L2 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 × 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 × fewer trainable parameters than both the monolithic SNN (red) and ANN (blue) baselines.
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.
Petr Badolia, Leonid Obukhov, Dmitry Bylinkin +1
Basic Research of Artificial Intelligence Laboratory (BRAIn Lab) · Innopolis University
Partial differential equations (PDEs) govern diverse physical phenomena, yet high-fidelity numerical solutions are computationally expensive and Machine Learning approaches lack generalization. While Scientific Foundation Models (SFMs) aim to provide universal surrogates, typical encoding-decoding approaches suffer from high pretraining costs and limited interpretability. In this paper, we propose Compositional Neural Operators (CompNO) for 2D systems, a framework that decomposes complex PDEs into a library of Foundation Blocks. Each block is a specialized Neural Operator pretrained on elementary physics. This modular library contains convection, diffusion, and nonlinear convection blocks as well as a Poisson Solver, enabling the framework to address the pressure-velocity coupling. These experts are assembled via an Adaptation Block featuring an Aggregator. This aggregator learns nonlinear interactions by minimizing data loss and physics-based residuals driven from governing equations. The proposed approach has been evaluated on the Convection-Diffusion equation, the Burgers' equation, and the Incompressible Navier-Stokes equation. Our results demonstrate that learning from elementary operators significantly improves adaptability, enhances model interpretability and facilitates the reuse of pretrained blocks when adapting to new physical systems.
Hamda Hmida, Hsiu-Wen Chang, Youssef Mesri
Mines Paris - PSL University, Centre for Material Forming (CEMEF) · Mines Paris - PSL University, Centre for Robotics (CAOR)
Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work introduces a physics-informed training framework that addresses these limitations by decomposing PDEs using operator splitting methods, training separate neural operators to learn individual non-linear physical operators while approximating linear operators with fixed finite-difference convolutions. This modular mixture-of-experts architecture enables generalisation to novel physical regimes by explicitly encoding the underlying operator structure. We formulate the modelling task as a neural ordinary differential equation (ODE) where these learned operators constitute the right-hand side, enabling continuous-in-time predictions through standard ODE solvers and implicitly enforcing PDE constraints. Demonstrated on incompressible and compressible Navier--Stokes equations, our approach achieves better convergence and superior performance when generalising to unseen physics. The method remains parameter-efficient, enabling temporal extrapolation beyond training horizons, and provides interpretable components whose behaviour can be verified against known physics.
Vignesh Gopakumar, Ander Gray, Dan Giles +5
UCL Centre for Artificial Intelligence · UK Atomic Energy Authority · LIX, CNRS, ´Ecole Polytechnique +2