mlsanityData to 7 Oct 2026

LNO: Laplace Neural Operator for Solving Differential Equations

arXiv 2303.10528 ↗First posted 19 Mar 2023cs.LGcited by 47

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Abstract (arXiv)

We introduce the Laplace neural operator (LNO), which leverages the Laplace transform to decompose the input space. Unlike the Fourier Neural Operator (FNO), LNO can handle non-periodic signals, account for transient responses, and exhibit exponential convergence. LNO incorporates the pole-residue relationship between the input and the output space, enabling greater interpretability and improved generalization ability. Herein, we demonstrate the superior approximation accuracy of a single Laplace layer in LNO over four Fourier modules in FNO in approximating the solutions of three ODEs (Duffing oscillator, driven gravity pendulum, and Lorenz system) and three PDEs (Euler-Bernoulli beam, diffusion equation, and reaction-diffusion system). Notably, LNO outperforms FNO in capturing transient responses in undamped scenarios. For the linear Euler-Bernoulli beam and diffusion equation, LNO's exact representation of the pole-residue formulation yields significantly better results than FNO. For the nonlinear reaction-diffusion system, LNO's errors are smaller than those of FNO, demonstrating the effectiveness of using system poles and residues as network parameters for operator learning. Overall, our results suggest that LNO represents a promising new approach for learning neural operators that map functions between infinite-dimensional spaces.

Key facts. Cited by 47 papers in the map (data through 7 Oct 2026). Builds on 3 of its 10 references that are in the map. First posted on arXiv 19 Mar 2023.

Builds on

3 of 10 references.

References · 3

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 14; 2024: 15; 2025: 8; 2026: 10.

47 papers.

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  1. 2024CViT: Continuous Vision Transformer for Operator Learningcited by 47
  2. 2023Nonlocality and Nonlinearity Implies Universality in Operator Learningcited by 43
  3. 2023CFDBench: A Large-Scale Benchmark for Machine Learning Methods in Fluid Dynamicscited by 40
  4. 2023Rethinking materials simulations: Blending direct numerical simulations with neural operatorscited by 25
  5. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  6. 2023Learning in latent spaces improves the predictive accuracy of deep neural operatorscited by 19
  7. 2023Real-time Inference and Extrapolation via a Diffusion-inspired Temporal Transformer Operator (DiTTO)cited by 16
  8. 2025Recurrent Neural Operators: Stable Long-Term PDE Predictioncited by 15
  9. 2024DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domainscited by 14
  10. 2024Separable DeepONet: Breaking the Curse of Dimensionality in Physics-Informed Machine Learningcited by 14
  11. 2024Derivative-informed neural operator acceleration of geometric MCMC for infinite-dimensional Bayesian inverse problemscited by 13
  12. 2023Sound propagation in realistic interactive 3D scenes with parameterized sources using deep neural operatorscited by 12
35 more
  1. 2024Generating synthetic data for neural operatorscited by 11
  2. 2025Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learningcited by 11
  3. 2024A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformationscited by 10
  4. 2024Accelerating Phase Field Simulations Through a Hybrid Adaptive Fourier Neural Operator with U-Net Backbonecited by 10
  5. 2023Neural oscillators for generalization of physics-informed machine learningcited by 9
  6. 2024Time-SSM: Simplifying and Unifying State Space Models for Time Series Forecastingcited by 9
  7. 2023MyCrunchGPT: A chatGPT assisted framework for scientific machine learningcited by 8
  8. 2024LazyDINO: Fast, scalable, and efficiently amortized Bayesian inversion via structure-exploiting and surrogate-driven measure transportcited by 7
  9. 2025On the definition and importance of interpretability in scientific machine learningcited by 5
  10. 2023Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problemscited by 4
  11. 2024Uncertainty quantification for deeponets with ensemble kalman inversioncited by 4
  12. 2024LES-SINDy: Laplace-Enhanced Sparse Identification of Nonlinear Dynamical Systemscited by 4
  13. 2025FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulationcited by 4
  14. 2023Capturing the Diffusive Behavior of the Multiscale Linear Transport Equations by Asymptotic-Preserving Convolutional DeepONetscited by 3
  15. 2023How important are specialized transforms in Neural Operators?cited by 2
  16. 2024Ricci flow regularization in latent spaces for the forward learning of partial differential equationscited by 2
  17. 2026Learning Data-Efficient and Generalizable Neural Operators via Fundamental Physics Knowledgecited by 2
  18. 2023DON-LSTM: Multi-Resolution Learning with DeepONets and Long Short-Term Memory Neural Networkscited by 1
  19. 2024A novel data generation scheme for surrogate modelling with deep operator networkscited by 1
  20. 2024Harnessing physics-informed operators for high-dimensional reliability analysis problemscited by 1
  21. 2025StFT: Spatio-temporal Fourier Transformer for Long-term Dynamics Predictioncited by 1
  22. 2025Extrapolating Phase-Field Simulations in Space and Time with Purely Convolutional Architecturescited by 1
  23. 2026Learning Gradient Flow: Using Equation Discovery to Accelerate Engineering Optimizationcited by 1
  24. 2026WLNO: Wavelet-Laplace Neural Operator for Solving Partial Differential Equationscited by 1
  25. 2023Machine learning for advancing low-temperature plasma modeling and simulationcited by 0
  26. 2024A novel framework for generalization of deep hidden physics modelscited by 0
  27. 2025MeshONet: A Generalizable and Efficient Operator Learning Method for Structured Mesh Generationcited by 0
  28. 2025From Local Interactions to Global Operators: Scalable Gaussian Process Operator for Physical Systemscited by 0
  29. 2026SFO: Learning PDE Operators via Spectral Filteringcited by 0
  30. 2026Physics-Informed Generative Solver: Bridging Data-Driven Priors and Conservation Laws for Stable Spatiotemporal Field Reconstructioncited by 0
  31. 2026Let There Be Light: Reflection, Refraction and Scattering for Neural Operatorscited by 0
  32. 2026GENERIC-FNO: Embedding Energy Conservation and Entropy Production into Fourier Neural Operatorscited by 0
  33. 2026From inverse problems to neural operators: prediction, mechanism, and generalization of data-driven modelscited by 0
  34. 2026Quantum SEDONet: Spectrally-Embedded Quantum Deep Operator Networks for Partial Differential Equationscited by 0
  35. 2026Nonlinear elliptic homogenization with the parametric Deep Ritz methodcited by 0