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LFNO: Bridging Laplace and Fourier via Transient-Steady Decomposition

The paper introduces the Laplace-Fourier Neural Operator (LFNO), a unified framework that decomposes dynamical systems into transient and steady-state components to significantly outperform existing operators on ODEs while maintaining competitive performance on PDEs across multiple temporal scales.

Original authors: Jeongun Ha, Sanga Yoon, Donghun Lee

Published 2026-06-09

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CC BY 4.0

Original authors: Jeongun Ha, Sanga Yoon, Donghun Lee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a computer to predict how a complex system—like a swinging pendulum, a flowing river, or a vibrating bridge—changes over time.

For a long time, scientists have used two main "languages" to teach computers this task:

The Fourier Language (FNO): This is great at describing things that repeat in a steady rhythm, like a heartbeat or a steady ocean wave. It's like listening to a song and identifying the notes that keep playing over and over.
The Laplace Language (LNO): This is better at describing things that happen once and then fade away, like a bell ringing and slowly silencing, or a car braking to a stop. It focuses on the "transient" moments—the start, the crash, the decay.

The Problem:
Existing models usually pick one language or the other.

If you use the Fourier model for a system that starts chaotic and then settles down, it gets confused by the "noise" of the start.
If you use the Laplace model for a complex, repeating system, it struggles to capture the intricate details of the repeating patterns.

The Solution: LFNO (The "Bilingual" Translator)
The authors of this paper created a new model called LFNO (Laplace–Fourier Neural Operator). Think of LFNO as a dual-brain system that splits the problem into two distinct tasks, solves them separately, and then combines the answers.

Here is how it works, using simple analogies:

1. The "Dual-Branch" Architecture

Imagine a construction crew building a house. Instead of having one crew try to lay the foundation and paint the walls at the same time, they split the work:

Branch A (The Transient Team): This team uses the Laplace method. Their job is to handle the "chaos at the start." They focus on the sudden changes, the rapid decays, and the initial jolts (like the shock of a car crash or the initial swing of a pendulum). They are experts at things that fade away.
Branch B (The Steady Team): This team uses the Fourier method. Their job is to handle the "steady rhythm." They focus on the repeating patterns and the long-term behavior (like the steady hum of an engine or the regular waves of the ocean). They are experts at things that keep going.

2. The "Mixing Bowl"

Once both teams finish their specific jobs, LFNO mixes their results together.

The Transient Team says: "Here is how the system reacts to the sudden shock."
The Steady Team says: "Here is how the system settles into its regular pattern."
LFNO adds them together to give you the full, accurate picture of the entire timeline.

3. Why This Matters (The Results)

The authors tested this "dual-brain" approach on nine different challenges, ranging from simple swinging pendulums (ODEs) to complex fluid dynamics like wind and water flow (PDEs).

On "Start-and-Stop" Systems (ODEs): When the system is dominated by sudden changes and decays (like a chaotic pendulum), LFNO crushed the competition. It was much better at predicting the "messy" start than the old models.
On "Steady" Systems (PDEs): Even for systems that are mostly steady (like heat spreading or fluid flowing), LFNO was just as good as the best existing models, and in some very difficult cases (like turbulent water with low viscosity), it was significantly better. It managed to predict the tiny, chaotic swirls in the water that other models missed.
Stability: The paper notes that LFNO is more stable. It doesn't get "confused" or make wild errors when the math gets tough. It's like a driver who keeps their cool in a storm, whereas other models might swerve off the road.

The Bottom Line

The paper claims that by explicitly separating the "sudden, fading" parts of a problem from the "steady, repeating" parts, LFNO creates a more robust and accurate tool for predicting how the physical world changes over time. It doesn't try to force one tool to do everything; instead, it uses the right tool for the right job and combines them perfectly.

Technical Summary: LFNO – Bridging Laplace and Fourier via Transient-Steady Decomposition

1. Problem Statement

Existing neural operator frameworks for solving differential equations face a fundamental trade-off when modeling dynamical systems that exhibit both transient (decaying) and steady-state (oscillatory) behaviors.

Fourier Neural Operators (FNO) excel at capturing complex, nonlinear modal interactions in the frequency domain and are robust for periodic steady-state problems. However, they rely on periodic boundary conditions and struggle with non-periodic, exponentially decaying transient dynamics.
Laplace Neural Operators (LNO) utilize a pole-residue formulation in the Laplace domain to naturally capture transient behaviors and non-periodic responses. However, the original LNO framework may suffer from limited expressivity in modeling complex, nonlinear dependencies within the steady-state component and can encounter optimization instability in stiff systems.

The core challenge is to develop a unified framework that retains the spectral advantages of Fourier analysis for steady-state components while leveraging the pole-residue capabilities of Laplace analysis for transient dynamics, without sacrificing stability or accuracy.

2. Methodology: The Laplace–Fourier Neural Operator (LFNO)

The authors propose LFNO, a unified framework that integrates the theoretical foundations of LNO and FNO through a dual-branch architecture that explicitly decomposes system dynamics into transient and steady-state components.

Mathematical Formulation:
The method models the dynamical system by substituting the kernel integral operator with a representation defined in the Laplace domain. Using a pole-residue formulation, the output $U(s)$ is decomposed into:

Transient Response: Associated with system poles $\mu_n$ , representing decaying dynamics.
Steady-State Response: Associated with the excitation poles $i\omega_\ell$ of the input signal, representing oscillatory dynamics.

The time-domain solution $u(t)$ is reconstructed as a superposition:
$u(t) = \sum_{n=1}^{N} \gamma_n e^{\mu_n t} + \sum_{\ell=-L}^{L} \lambda_\ell e^{i\omega_\ell t}$

Architectural Integration:

Transient Branch: Operates in the time-domain (via the inverse Laplace transform of the pole-residue terms) to model decaying exponential dynamics. The authors allocate more layers to this module, motivated by the observation that transient phases often involve rapid transitions and complex gradients requiring higher model capacity.
Steady-State Branch: Operates in the frequency domain. Instead of relying solely on linear pole-residue coefficients, LFNO applies a learned Fourier integral operator (similar to FNO) to the steady-state coefficients. A learnable weight matrix $R_\ell$ updates the coefficients $\lambda_\ell$ for each frequency mode, allowing the model to capture complex nonlinear spectral couplings that simple linear methods might miss.
Synthesis: The final solution is the additive superposition of the preserved transient dynamics and the filtered, enhanced steady-state response.

3. Key Contributions

Unified Framework: Development of LFNO, which theoretically unifies Laplace-domain pole-residue modeling with Fourier spectral analysis.
Dual-Branch Architecture: Introduction of an architecture-level integration that processes transient and steady-state components separately, ensuring high-fidelity approximation of complex temporal evolutions.
Empirical Validation: Comprehensive benchmarking demonstrating that LFNO significantly outperforms existing operators (LNO and FNO) on Ordinary Differential Equations (ODEs) dominated by transient behavior, while maintaining competitive performance with FNO on Partial Differential Equation (PDE) benchmarks.

4. Experimental Results

The authors evaluated LFNO on nine benchmarks: three ODE systems (Duffing, Lorenz, Pendulum) and six PDE systems (Euler–Bernoulli beam, Heat, Reaction–diffusion, Brusselator, Burgers, and Navier–Stokes).

ODE Performance: LFNO significantly outperformed both FNO and LNO on ODE tasks, particularly in systems with transient dynamics (e.g., Lorenz $\rho=10$ , Duffing $c=0.5$ ). It achieved lower $L_2$ and $L_\infty$ errors, demonstrating superior robustness in handling complex nonlinearities and long-term damping.
PDE Performance: LFNO consistently surpassed LNO across all PDE benchmarks. Compared to FNO, LFNO achieved competitive or superior results, with notable improvements in high-complexity regimes:
- Burgers' Equation: LFNO reduced the $L_\infty$ error to 0.0125, nearly a four-fold reduction compared to FNO (0.0471), effectively mitigating spectral ringing artifacts near shock fronts.
- Navier–Stokes (Low Viscosity): In the challenging $\nu = 10^{-3}$ regime, LFNO achieved an $L_2$ error of 0.0026, outperforming FNO (0.0286) by over an order of magnitude.
Stability and Convergence: Training curves indicated that LFNO maintained monotonic and stable loss decay, whereas FNO frequently exhibited stagnation in chaotic regimes and LNO showed significant fluctuations. The authors attribute this to the transient-steady decomposition acting as an implicit regularizer, preventing overfitting to high-frequency spectral noise.

5. Significance and Claims

The paper claims that LFNO provides a robust and unified approach for learning complex dynamical systems across multiple temporal scales. By explicitly bridging the Laplace and Fourier domains, LFNO resolves the inherent trade-offs between modeling transient decay and steady-state oscillation.

The authors emphasize that their results demonstrate:

Improved Stability: The component-wise decomposition leads to a more stable loss landscape, preventing the model from getting trapped in suboptimal local minima.
Physical Interpretability: The separation of dynamics offers a more interpretable learning process, particularly for stiff systems characterized by multi-scale spatiotemporal features.
Data Efficiency: LFNO demonstrated superior sample efficiency, achieving high accuracy even with a restricted sample size (380 trajectories) across diverse benchmarks.

The work suggests that the success of this hybrid approach points toward a broader potential for hybrid basis expansion in neural operator learning, potentially extending to other orthogonal bases (e.g., Wavelets, Chebyshev polynomials) for even more complex, non-homogeneous spatiotemporal dynamics. The authors position LFNO not just as a specific solution for transient-steady separation, but as a foundation for adaptive basis learning in high-dimensional physical systems.