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arxiv: 2606.07601 · v1 · pith:GCB3LYGXnew · submitted 2026-05-29 · 💻 cs.LG · cs.AI

LFNO: Bridging Laplace and Fourier via Transient-Steady Decomposition

Jeongun Ha , Sanga Yoon , Donghun Lee This is my paper

Pith reviewed 2026-06-28 23:22 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords Laplace-Fourier Neural Operatortransient-steady decompositionneural operatorsdynamical systemsODE modelingPDE benchmarksspectral methods
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The pith

LFNO decomposes dynamical responses into transient and steady-state parts using separate Laplace and Fourier branches.

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

The paper introduces LFNO as a neural operator that splits modeling of time-evolving systems into two independent paths: a Laplace branch for the initial changing phase and a Fourier branch for the long-term settled behavior. This split is tested on ODE examples like Duffing, Lorenz, and pendulum, where rapid shifts matter most, and on PDE examples like heat flow, Burgers equation, and Navier-Stokes. The approach yields stronger accuracy than prior operators when transients dominate, stays competitive with pure Fourier methods on PDE tasks, and produces more stable training with clearer physical meaning through the separated components. If the split works as claimed, models can respect distinct time scales without one spectral tool handling every regime.

Core claim

LFNO employs a dual-branch architecture that explicitly decomposes system dynamics into transient and steady-state components, integrating the spectral advantages of Laplace and Fourier Neural Operators to provide a unified framework for learning complex dynamical systems across multiple temporal scales, with demonstrated gains on ODE benchmarks and competitive PDE results plus improved stability and interpretability.

What carries the argument

Dual-branch architecture that decomposes responses into transient (Laplace) and steady-state (Fourier) components.

If this is right

  • LFNO outperforms prior operators on ODE systems where transient dynamics dominate.
  • It surpasses LNO and matches FNO on PDE benchmarks.
  • Component-wise decomposition yields improved stability and physical interpretability.

Where Pith is reading between the lines

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

  • The separation could let practitioners update only the transient branch when initial conditions change while keeping the steady branch fixed.
  • Similar dual-branch ideas might apply to other paired transforms for systems with multiple distinct scales.

Load-bearing premise

Dynamical system responses can be cleanly decomposed into independent transient and steady-state components best captured by separate Laplace and Fourier branches without significant interaction or information loss.

What would settle it

Performance drop or training instability on a coupled transient-steady system when the model is forced to use the two-branch split versus a single unified branch.

Figures

Figures reproduced from arXiv: 2606.07601 by Donghun Lee, Jeongun Ha, Sanga Yoon.

Figure 1
Figure 1. Figure 1: Model architecture of LFNO. (a) Overall process: given input function f(t), 1. embed the input function f(t) into a higher￾dimensional latent space v(t) via a shallow neural network P, 2. apply Laplace transform L, 3. take the Transient and the Steady layers for each component, and 3. Project the integrated result back to the target space to obtain the output u(t). (b) Steady layer structure: Hybrid spectr… view at source ↗
Figure 2
Figure 2. Figure 2: Qualitative comparison on six PDE benchmarks. Columns correspond to ground truth (GT), FNO, LFNO, and LNO predictions. Full-resolution results are provided in Section E. all PDE benchmarks. Most notably, in the Burgers’ equation, LFNO yields an L∞ error of 0.0125, representing nearly a four-fold reduction compared to FNO’s 0.0471. This substantial suppression of the maximum point-wise error proves that LFN… view at source ↗
Figure 4
Figure 4. Figure 4: PDE learning curves on different models. Training and validation sets are represented by lime and red for LFNO, orange and blue for LNO, and magenta and cyan for FNO. target mapping consistency within the datasets. In the orig￾inal LNO datasets, we observed that the forcing functions used to generate training samples differed from those used for validation. From a functional analysis perspective, this impl… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of point-wise Mean Absolute Error (MAE) between LNO and LFNO across ODE systems. The time horizon is partitioned into the transient state (t < 500), and the steady-state (t > 500). (a), (b) Duffing (c = 0, 0.5), (c), (d) Lorenz (ρ = 5, 10), and (e) Pendulum (c = 0.5). The green lines represent the LNO, while the magenta lines denote the LFNO. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Euler-Bernoulli beam equation: Comparison of qualitative results computed by FNO, LFNO, and LNO. It represents the ground truth shown in the left plot, the prediction of each model in the center plot, and the (truth, pred) scatters are shown in the right plot. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Heat equation: Comparison of qualitative results computed by FNO, LFNO, and LNO. It represents the ground truth shown in the left plot, the prediction of each model in the center plot, and the (truth, pred) scatters are shown in the right plot. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Reaction-diffusion equation: Comparison of qualitative results computed by FNO, LFNO, and LNO. It represents the ground truth shown in the left plot, the prediction of each model in the center plot, and the (truth, pred) scatters are shown in the right plot. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Brusselator equation: Comparison of qualitative results computed by FNO, LFNO, and LNO. It represents the ground truth shown in the left plot, the prediction of each model in the center plot, and the (truth, pred) scatters are shown in the right plot. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Burgers’ equation: Comparison of qualitative results computed by FNO, LFNO, and LNO. It represents the ground truth shown in the left plot, the prediction of each model in the center plot, and the (truth, pred) scatters are shown in the right plot. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Navier-Stokes equation: Comparison of qualitative results computed by FNO, LFNO, and LNO. It represents the ground truth shown in the left plot, the prediction of each model in the center plot, and the (truth, pred) scatters are shown in the right plot. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
read the original abstract

We introduce the Laplace-Fourier Neural Operator (LFNO), a unified framework for modeling dynamical systems across transient and steady-state regimes by integrating the spectral advantages of Laplace and Fourier Neural Operators. LFNO employs a dual-branch architecture that explicitly decomposes system dynamics into transient and steady-state components. We evaluate LFNO on nine benchmarks, including three ODE systems (Duffing, Lorenz, and Pendulum) and six PDE systems (Euler-Bernoulli beam, Heat, Reaction-diffusion, Brusselator, Burgers, and Navier-Stokes). LFNO significantly outperforms existing operators on ODE systems, where transient dynamics dominate, and consistently surpasses LNO while achieving performance competitive with FNO on PDE benchmarks. Furthermore, LFNO offers improved stability and physical interpretability through its component-wise decomposition. These results demonstrate that LFNO provides a robust and unified approach for learning complex dynamical systems across multiple temporal scales.

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 / 2 minor

Summary. The paper introduces LFNO, a dual-branch neural operator that decomposes dynamical system responses into transient components (modeled via a Laplace branch) and steady-state components (modeled via a Fourier branch). It evaluates the method on three ODE benchmarks (Duffing, Lorenz, Pendulum) and six PDE benchmarks (Euler-Bernoulli beam, Heat, Reaction-diffusion, Brusselator, Burgers, Navier-Stokes), claiming significant outperformance over existing operators on ODEs, consistent improvement over LNO, and competitive results with FNO on PDEs, along with gains in stability and physical interpretability from the explicit decomposition.

Significance. If the decomposition mechanism is shown to be responsible for the gains rather than capacity alone, LFNO would provide a principled way to combine the complementary strengths of Laplace and Fourier transforms for multi-regime dynamics, extending neural operators to systems with dominant transients while retaining PDE performance. The explicit component-wise structure could also support interpretability in learned operators.

major comments (2)
  1. [Experiments] Experiments (likely §4 or §5): The reported outperformance on the three ODE systems is presented as evidence for the benefit of assigning Laplace to transients and Fourier to steady-state, but no ablation is described that compares LFNO against a same-capacity dual-branch baseline in which both branches use the same transform (e.g., two Fourier branches or two Laplace branches). Without this control, it is impossible to determine whether the gains arise from the specific transient-steady assignment or simply from the added parallel capacity.
  2. [§3] §3 (Architecture): The claim that the dual-branch design enforces a clean separation without significant cross-regime interaction or information loss is central to the interpretability and stability arguments, yet the manuscript provides no quantitative verification (e.g., cross-branch leakage metrics, reconstruction error when one branch is ablated, or visualization of learned component isolation) on the ODE test cases where the separation is asserted to be most beneficial.
minor comments (2)
  1. [§3] Notation for the transient and steady-state projections (likely Eq. (3)–(5)) could be clarified to make explicit whether the decomposition is learned jointly or imposed via separate loss terms.
  2. [Figures] Figure captions for the benchmark results should include error bars or standard deviations across random seeds to allow assessment of statistical significance of the reported improvements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback. The two major comments highlight important gaps in experimental validation that we will address through targeted additions in the revised manuscript.

read point-by-point responses

  1. Referee: [Experiments] Experiments (likely §4 or §5): The reported outperformance on the three ODE systems is presented as evidence for the benefit of assigning Laplace to transients and Fourier to steady-state, but no ablation is described that compares LFNO against a same-capacity dual-branch baseline in which both branches use the same transform (e.g., two Fourier branches or two Laplace branches). Without this control, it is impossible to determine whether the gains arise from the specific transient-steady assignment or simply from the added parallel capacity.

    Authors: We agree that the current experiments do not include a same-capacity dual-branch control with identical transforms in both branches, which would more directly isolate the contribution of the transient-steady assignment. In the revised version we will add these ablations on the three ODE benchmarks (Duffing, Lorenz, Pendulum), reporting results for (i) two Fourier branches and (ii) two Laplace branches, each with parameter counts matched to LFNO. This will allow readers to distinguish architecture capacity from the specific Laplace-Fourier decomposition. revision: yes

  2. Referee: [§3] §3 (Architecture): The claim that the dual-branch design enforces a clean separation without significant cross-regime interaction or information loss is central to the interpretability and stability arguments, yet the manuscript provides no quantitative verification (e.g., cross-branch leakage metrics, reconstruction error when one branch is ablated, or visualization of learned component isolation) on the ODE test cases where the separation is asserted to be most beneficial.

    Authors: We acknowledge that the manuscript currently lacks quantitative verification of branch separation on the ODE cases. In the revision we will add (i) visualizations of the transient and steady-state components learned on the ODE test trajectories, (ii) reconstruction error when each branch is ablated individually, and (iii) a simple cross-branch leakage metric (e.g., L2 norm of the component that should be zero under ideal separation). These additions will be placed in §3 and §4 to support the interpretability claims. revision: yes

Circularity Check

0 steps flagged

No circularity: LFNO is an explicit dual-branch architectural proposal with external benchmark evaluation.

full rationale

The paper introduces LFNO as a new operator architecture that decomposes dynamics into transient (Laplace) and steady-state (Fourier) branches. No derivation step reduces a claimed prediction or uniqueness result to a fitted parameter or self-citation by construction. The central claims rest on empirical performance across nine standard benchmarks rather than any algebraic identity or imported theorem that collapses to the input data. Self-citations, if present, are not load-bearing for the decomposition mechanism itself. This is the normal case of an architectural contribution whose validity is tested externally.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Based solely on the abstract, the central claim rests on the domain assumption that clean separation of transient and steady-state dynamics is possible and advantageous; no free parameters or invented entities are explicitly quantified in the provided text.

axioms (1)
  • domain assumption Dynamical systems can be explicitly decomposed into transient and steady-state components that are independently and optimally modeled by Laplace and Fourier branches.
    This premise directly justifies the dual-branch design and the claimed gains in stability and interpretability.
invented entities (1)
  • LFNO dual-branch architecture no independent evidence
    purpose: To separately model transient and steady-state dynamics using Laplace and Fourier operators.
    New architecture introduced in the paper; no independent evidence outside the abstract is provided.

pith-pipeline@v0.9.1-grok · 5678 in / 1253 out tokens · 28778 ms · 2026-06-28T23:22:12.137244+00:00 · methodology

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