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arxiv: 2604.11625 · v1 · submitted 2026-04-13 · 💻 cs.LG · cs.AI

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

Samrendra Roy , Souvik Chakraborty , Rizwan-uddin , Syed Bahauddin Alam This is my paper

Pith reviewed 2026-05-10 15:10 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords spiking neural networksneural operatorscompositional architecturesPDE surrogatesmodular learningneuromorphic computingnuclear diffusion equations
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The pith

A library of small spiking blocks for elementary PDE operators composes via an aggregator and correction network to solve unseen coupled systems with far fewer parameters than monolithic models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that spiking neural operator blocks trained only on isolated terms such as convection or diffusion can be assembled through an input-conditioned aggregator to handle coupled PDEs never seen in block training. A lightweight correction network absorbs cross-term residuals while the original blocks stay frozen, enabling modular growth without catastrophic forgetting. This construction is evaluated across eight PDE families, including five coupled systems and a nuclear neutron diffusion equation, where the full SCNO pipeline records the lowest relative L2 error on four of the five coupled cases. The architecture uses only 95K trainable parameters against 462K in a monolithic spiking baseline and delivers error reductions up to 62 percent relative to that baseline. If the composition holds, the approach supplies an energy-efficient, incrementally expandable surrogate solver suited to neuromorphic hardware and to physics domains where new equations appear over time.

Core claim

SCNO maintains a library of small spiking neural operator blocks, each trained on a single elementary differential operator, 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.

What carries the argument

The modular composition of pre-trained spiking operator blocks through an input-conditioned aggregator together with a frozen residual-correction network.

If this is right

  • SCNO records the lowest relative L2 error on four of five coupled PDEs while using only 95K trainable parameters versus 462K for the monolithic spiking baseline.
  • Error reductions reach up to 62 percent against a monolithic spiking DeepONet and 65 percent against a standard ANN DeepONet across the tested coupled systems.
  • The frozen-block design guarantees that adding new elementary operators or new physics never overwrites performance on previously solved equations.
  • The same modular pipeline applies to a nuclear-relevant one-group neutron diffusion equation among the evaluated families.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If elementary blocks compose reliably, a growing library of such blocks could serve as the kernel of a foundation model for PDEs in which novel physics are accommodated by training only the new block and a fresh correction head.
  • The separation of operator blocks from the aggregator suggests that the same structure could be transferred to other operator-learning settings such as integral equations or stochastic PDEs without retraining the entire surrogate.
  • Direct measurement of spike-based energy cost on neuromorphic chips would quantify whether the accuracy gains also translate into the expected reduction in inference power.

Load-bearing premise

Blocks trained separately on single operators remain accurate when their outputs are aggregated and lightly corrected to represent coupled systems that were never shown during block training.

What would settle it

On any new coupled PDE the composed SCNO with correction produces a higher relative L2 error than a freshly trained monolithic DeepONet of comparable capacity.

Figures

Figures reproduced from arXiv: 2604.11625 by Rizwan-uddin, Samrendra Roy, Souvik Chakraborty, Syed Bahauddin Alam.

Figure 1
Figure 1. Figure 1: SCNO architecture. Frozen spiking operator blocks [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Training convergence for elementary spiking [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Representative predictions from elementary spiking blocks on held-out test samples. Each row corresponds to one [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Nuclear case study: 1-group neutron diffusion [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: Forgetting-free modular expansion: blocks are [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 5
Figure 5. Figure 5: Coupled PDE predictions across all five systems and four methods. Each row is a coupled PDE; columns show different [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison across five coupled PDEs: (left) relative [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Accuracy–efficiency Pareto analysis. SCNO+Corr [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims to introduce the Spiking Compositional Neural Operator (SCNO), a modular architecture combining spiking and conventional components. It maintains a library of small spiking neural operator blocks each trained on a single elementary operator (convection, diffusion, reaction), composes them via a lightweight input-conditioned aggregator to solve coupled PDEs unseen during block training, and uses a small correction network to learn cross-coupling residuals while keeping blocks and aggregator frozen. Evaluated on eight PDE families including five coupled systems and a nuclear 1-group neutron diffusion equation, SCNO with correction reports the lowest relative L² error on four of five coupled PDEs, outperforming a monolithic spiking DeepONet (by up to 62%, mean over 3 seeds) and standard ANN DeepONet (by up to 65%), with only 95K trainable parameters versus 462K for the monolithic baseline, and is positioned as the first compositional spiking neural operator enabling forgetting-free modular expansion.

Significance. If the compositional claims are substantiated, this could enable energy-efficient neuromorphic PDE solvers with reusable modular components and zero-forgetting expansion, which is relevant for nuclear applications. The reported parameter reduction and performance gains on held-out coupled systems, along with means over three seeds, are concrete strengths that would support broader adoption of spiking operators if the modularity holds without the correction network subsuming the design.

major comments (2)
  1. Abstract and evaluation on coupled PDEs: the central claim that blocks trained solely on isolated operators can be frozen and composed via the aggregator for unseen coupled systems (with only a small correction network) lacks ablations isolating the aggregator's contribution or the correction network's size and role relative to the blocks. This is load-bearing for the assertions of modularity, zero-forgetting expansion, and the parameter reduction (95K vs 462K), as the aggregator or correction could learn effective couplings by construction.
  2. Evaluation section: details on spiking block architecture, training procedure for blocks/aggregator/correction, and statistical tests for the reported error reductions (means over 3 seeds) are absent, which prevents verification of the soundness of the performance claims on the five coupled PDEs and the nuclear neutron diffusion equation.
minor comments (1)
  1. The abstract's claim of being 'the first' compositional spiking neural operator would benefit from explicit comparison to prior work on compositional neural operators in the related work section for context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments. We address each major point below and commit to revisions that enhance the manuscript's clarity and rigor.

read point-by-point responses

  1. Referee: Abstract and evaluation on coupled PDEs: the central claim that blocks trained solely on isolated operators can be frozen and composed via the aggregator for unseen coupled systems (with only a small correction network) lacks ablations isolating the aggregator's contribution or the correction network's size and role relative to the blocks. This is load-bearing for the assertions of modularity, zero-forgetting expansion, and the parameter reduction (95K vs 462K), as the aggregator or correction could learn effective couplings by construction.

    Authors: We agree that targeted ablations would more convincingly isolate the aggregator's role and the correction network's contribution. In the revised manuscript we will add: (i) performance comparisons of the frozen blocks plus aggregator with and without the correction network, (ii) an ablation disabling the input-conditioned aggregator (replacing it with a fixed linear combination), and (iii) explicit parameter breakdowns showing that the reported 95K trainable parameters are only for the aggregator plus correction network while the spiking blocks remain frozen and pre-trained. These additions will directly address whether the correction network subsumes the compositional design and will reinforce the zero-forgetting property, which holds by construction because block weights are never updated after pre-training. revision: yes

  2. Referee: Evaluation section: details on spiking block architecture, training procedure for blocks/aggregator/correction, and statistical tests for the reported error reductions (means over 3 seeds) are absent, which prevents verification of the soundness of the performance claims on the five coupled PDEs and the nuclear neutron diffusion equation.

    Authors: We acknowledge that the current evaluation section is insufficiently detailed for full reproducibility and verification. The revised version will expand this section to include: (1) the exact spiking block architecture (neuron model, layer counts, spike encoding, and block sizes), (2) complete training protocols for the blocks (on isolated operators), the aggregator, and the correction network (loss functions, optimizers, learning rates, epochs, and data splits), and (3) statistical reporting beyond means, specifically standard deviations across the three seeds together with any paired statistical tests used to support the reported error reductions. These details will be placed in the main text where space permits and otherwise moved to a new appendix subsection. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical claims rest on held-out evaluations

full rationale

The paper presents no mathematical derivation chain or first-principles predictions. Its headline results are direct empirical measurements of relative L2 error on five coupled PDE families never seen during block training, compared against monolithic spiking and ANN DeepONet baselines. The phrase 'preserving zero-forgetting modular expansion by construction' is an explicit design statement arising from the decision to freeze pre-trained blocks and the aggregator; it does not derive a new quantity from fitted parameters or reduce to a self-referential definition. No equations, uniqueness theorems, or self-citations are invoked to support the central performance claims. The evaluation protocol (held-out PDEs, parameter counts, seed-averaged improvements) is independent of the architectural description and does not collapse to quantities defined from the same fitted values.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 2 invented entities

The central claim rests on standard neural network approximation assumptions plus the novel compositional mechanism; no explicit free parameters beyond typical network hyperparameters are stated in the abstract.

free parameters (1)
  • neural network hyperparameters and block sizes
    Typical for any neural operator; specific values not reported in abstract but required to achieve the stated parameter count of 95K.
axioms (1)
  • domain assumption Spiking neural networks can serve as effective surrogates for elementary differential operators when trained on single-term PDEs
    Invoked when assigning separate blocks to convection, diffusion, and reaction.
invented entities (2)
  • Spiking Compositional Neural Operator (SCNO) no independent evidence
    purpose: Modular library-based PDE solver with zero-forgetting expansion
    New architecture combining spiking blocks, aggregator, and correction network.
  • input-conditioned aggregator no independent evidence
    purpose: Selects and combines frozen spiking blocks for new coupled PDEs
    Lightweight component introduced to enable composition.

pith-pipeline@v0.9.0 · 5569 in / 1449 out tokens · 71611 ms · 2026-05-10T15:10:44.185544+00:00 · methodology

discussion (0)

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Reference graph

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