LNO: Laplace Neural Operator for Solving Differential Equations

• URL: http://arxiv.org/abs/2303.10528v2
• Date: Tue, 30 May 2023 16:17:49 GMT
• Title: LNO: Laplace Neural Operator for Solving Differential Equations
• Abstract summary: We introduce the Laplace neural operator (LNO), which leverages the Laplace transform to decompose the input space. LNO incorporates the pole-residue relationship between the input and the output space, enabling greater interpretability. We show that LNO represents a promising new approach for learning neural operators that map functions between infinite-dimensional spaces.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: 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.

Related papers

• Linearized PINN with pretrained nonlinear layers [48.01541150431857]
  lPINN is a reduced-order neural basis method for forward and inverse differential equations.<n>We evaluate lPINN on forward and inverse problems for the advection-diffusion equation, Burgers' equation, and the nonlinear pendulum equation.
  arXiv  Detail & Related papers  (2026-09-14T02:18:00Z)
• WLNO: Wavelet-Laplace Neural Operator for Solving Partial Differential Equations [0.6008132390640294]
  Wavelet-Laplace Neural Operator fuses Haar wavelet multi-scale spatial decomposition with the Laplace-domain pole-residue formulation of the Laplace Neural Operator (LNO)<n>LNO captures transient and steady-state dynamics through learnable system poles and residues, but lacks an explicit mechanism for extracting localized multi-scale features inherent in complex PDE solutions.<n> WLNO addresses this by augmenting the LNO core with a parallel single-level Haar discrete wavelet transform (DWT) branch that decomposes the lifted feature map into four frequency subbands.<n> WLNO consistently outperforms LNO on
  arXiv  Detail & Related papers  (2026-05-23T16:58:37Z)
• FNO$^{\angle θ}$: Extended Fourier neural operator for learning state and optimal control of distributed parameter systems [1.1602089225841632]
  We show that any state and optimal control of linear PDEs with constant coefficients can be represented as an integral in the complex domain.<n>We illustrate the performance of FNO in learning state and optimal control for the nonlinear Burgers' equation.
  arXiv  Detail & Related papers  (2026-04-06T21:33:30Z)
• Linear-Nonlinear Fusion Neural Operator for Partial Differential Equations [4.892307363245757]
  We find that explicitly decoupling linear and nonlinear effects within operator mappings leads to markedly improved learning efficiency.<n>This yields a novel network structure, namely the Linear-Nonlinear Fusion Neural Operator (LNF-NO)<n>LNF-NO naturally supports multiple functional inputs and is applicable to both regular grids and irregular geometries.
  arXiv  Detail & Related papers  (2026-03-25T10:10:53Z)
• Derivative-Informed Fourier Neural Operator: Universal Approximation and Applications to PDE-Constrained Optimization [0.9236074230806578]
  We present approximation theories and efficient training methods for derivative-informed Fourier neural operators (DIFNOs)<n>A DIFNO is trained by minimizing its prediction error jointly on output and Fréchet derivative samples of a high-fidelity operator.<n>We show that DIFNOs are superior in sample complexity for operator learning and solving infinite-dimensional PDE-constrained inverse problems.
  arXiv  Detail & Related papers  (2025-12-16T04:54:24Z)
• Sumudu Neural Operator for ODEs and PDEs [2.497936211748472]
  We introduce the Sumudu Neural Operator (SNO), a neural operator rooted in the properties of the Sumudu Transform.<n>SNO achieves superior performance to FNO on PDEs and demonstrates competitive accuracy with LNO on several PDE tasks, including the lowest error on the Euler-Bernoulli Beam and Diffusion Equation.<n>These preliminary findings suggest promise for the Sumudu Transform as a neural operator design, particularly for certain classes of PDEs.
  arXiv  Detail & Related papers  (2025-11-14T02:57:03Z)
• Conservative approximation-based feedforward neural network for WENO schemes [4.867849275247251]
  We present the feedforward neural network based on the conservative approximation to the derivative from point values.<n>We present WENO3-CADNNs, where they outperform WENO3-Z and achieve accuracy comparable to WENO5-JS.
  arXiv  Detail & Related papers  (2025-07-08T17:19:48Z)
• Solving Poisson Equations using Neural Walk-on-Spheres [80.1675792181381]
  We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. We demonstrate the superiority of NWoS in accuracy, speed, and computational costs.
  arXiv  Detail & Related papers  (2024-06-05T17:59:22Z)
• On the locality of local neural operator in learning fluid dynamics [7.773086303468703]
  Local neural operator (LNO) is the core that enables LNO great flexibility in solving transient partial differential equations (PDEs) We investigate the locality of LNO by looking into its receptive field and receptive range. We deem rules found in this paper general when applying LNO to learn and solve transient PDEs in diverse fields.
  arXiv  Detail & Related papers  (2023-12-15T14:18:54Z)
• Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEs [93.82811501035569]
  We introduce a new data efficient and highly parallelizable operator learning approach with reduced memory requirement and better generalization. MG-TFNO scales to large resolutions by leveraging local and global structures of full-scale, real-world phenomena. We demonstrate superior performance on the turbulent Navier-Stokes equations where we achieve less than half the error with over 150x compression.
  arXiv  Detail & Related papers  (2023-09-29T20:18:52Z)
• Towards Large-Scale Learned Solvers for Parametric PDEs with Model-Parallel Fourier Neural Operators [3.0384874162715856]
  Fourier neural operators (FNOs) are a recently introduced neural network architecture for learning solution operators of partial differential equations. We propose a model-parallel version of FNOs based on domain-decomposition of both the input data and network weights. We demonstrate that our model-parallel FNO is able to predict time-varying PDE solutions of over 3.2 billions variables.
  arXiv  Detail & Related papers  (2022-04-04T02:12:03Z)
• Local neural operator for solving transient partial differential equations on varied domains [8.905324065830861]
  We propose local neural operator (LNO) for solving transient partial differential equations (PDEs) on varied domains. It comes together with a handy strategy including boundary treatments, enabling one pre-trained LNO to predict solutions on different domains. For demonstration, LNO learns Navier-Stokes equations from randomly generated data samples, and then the pre-trained LNO is used as an explicit numerical time-marching scheme.
  arXiv  Detail & Related papers  (2022-03-11T06:41:51Z)
• Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equations [14.43135909469058]
  We propose a novel textitpseudo-differential integral operator (PDIO) to analyze and generalize the Fourier integral operator in FNO. We experimentally validate the effectiveness of the proposed model by utilizing Darcy flow and the Navier-Stokes equation.
  arXiv  Detail & Related papers  (2022-01-28T07:22:32Z)
• Factorized Fourier Neural Operators [77.47313102926017]
  The Factorized Fourier Neural Operator (F-FNO) is a learning-based method for simulating partial differential equations. We show that our model maintains an error rate of 2% while still running an order of magnitude faster than a numerical solver.
  arXiv  Detail & Related papers  (2021-11-27T03:34:13Z)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)
• Incorporating NODE with Pre-trained Neural Differential Operator for Learning Dynamics [73.77459272878025]
  We propose to enhance the supervised signal in learning dynamics by pre-training a neural differential operator (NDO) NDO is pre-trained on a class of symbolic functions, and it learns the mapping between the trajectory samples of these functions to their derivatives. We provide theoretical guarantee on that the output of NDO can well approximate the ground truth derivatives by proper tuning the complexity of the library.
  arXiv  Detail & Related papers  (2021-06-08T08:04:47Z)
• Fourier Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  We formulate a new neural operator by parameterizing the integral kernel directly in Fourier space. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. It is up to three orders of magnitude faster compared to traditional PDE solvers.
  arXiv  Detail & Related papers  (2020-10-18T00:34:21Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.