# LNO: Laplace Neural Operator for Solving Differential Equations

arXiv: https://arxiv.org/abs/2303.10528 · HTML: https://mlsanity.com/paper/2303.10528/

## Key facts

- Cited by 47 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 10 references that are in the map.
- First posted on arXiv 19 Mar 2023.

## Builds on (most cited first, 3 of 3)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)

## Built on by (most cited first, 30 of 47)

- [CViT: Continuous Vision Transformer for Operator Learning](https://mlsanity.com/paper/2405.13998/) (2024)
- [CFDBench: A Large-Scale Benchmark for Machine Learning Methods in Fluid Dynamics](https://mlsanity.com/paper/2310.05963/) (2023)
- [Nonlocality and Nonlinearity Implies Universality in Operator Learning](https://mlsanity.com/paper/2304.13221/) (2023)
- [Rethinking materials simulations: Blending direct numerical simulations with neural operators](https://mlsanity.com/paper/2312.05410/) (2023)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Learning in latent spaces improves the predictive accuracy of deep neural operators](https://mlsanity.com/paper/2304.07599/) (2023)
- [Real-time Inference and Extrapolation via a Diffusion-inspired Temporal Transformer Operator (DiTTO)](https://mlsanity.com/paper/2307.09072/) (2023)
- [DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domains](https://mlsanity.com/paper/2402.07250/) (2024)
- [Recurrent Neural Operators: Stable Long-Term PDE Prediction](https://mlsanity.com/paper/2505.20721/) (2025)
- [Separable DeepONet: Breaking the Curse of Dimensionality in Physics-Informed Machine Learning](https://mlsanity.com/paper/2407.15887/) (2024)
- [Derivative-informed neural operator acceleration of geometric MCMC for infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2403.08220/) (2024)
- [Sound propagation in realistic interactive 3D scenes with parameterized sources using deep neural operators](https://mlsanity.com/paper/2308.05141/) (2023)
- [Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning](https://mlsanity.com/paper/2506.10973/) (2025)
- [Generating synthetic data for neural operators](https://mlsanity.com/paper/2401.02398/) (2024)
- [A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformations](https://mlsanity.com/paper/2404.00074/) (2024)
- [Accelerating Phase Field Simulations Through a Hybrid Adaptive Fourier Neural Operator with U-Net Backbone](https://mlsanity.com/paper/2406.17119/) (2024)
- [Neural oscillators for generalization of physics-informed machine learning](https://mlsanity.com/paper/2308.08989/) (2023)
- [Time-SSM: Simplifying and Unifying State Space Models for Time Series Forecasting](https://mlsanity.com/paper/2405.16312/) (2024)
- [MyCrunchGPT: A chatGPT assisted framework for scientific machine learning](https://mlsanity.com/paper/2306.15551/) (2023)
- [LazyDINO: Fast, scalable, and efficiently amortized Bayesian inversion via structure-exploiting and surrogate-driven measure transport](https://mlsanity.com/paper/2411.12726/) (2024)
- [On the definition and importance of interpretability in scientific machine learning](https://mlsanity.com/paper/2505.13510/) (2025)
- [FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulation](https://mlsanity.com/paper/2505.20349/) (2025)
- [Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems](https://mlsanity.com/paper/2307.00296/) (2023)
- [Uncertainty quantification for deeponets with ensemble kalman inversion](https://mlsanity.com/paper/2403.03444/) (2024)
- [LES-SINDy: Laplace-Enhanced Sparse Identification of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2411.01719/) (2024)
- [Capturing the Diffusive Behavior of the Multiscale Linear Transport Equations by Asymptotic-Preserving Convolutional DeepONets](https://mlsanity.com/paper/2306.15891/) (2023)
- [How important are specialized transforms in Neural Operators?](https://mlsanity.com/paper/2308.09293/) (2023)
- [Ricci flow regularization in latent spaces for the forward learning of partial differential equations](https://mlsanity.com/paper/2401.14591/) (2024)
- [Learning Data-Efficient and Generalizable Neural Operators via Fundamental Physics Knowledge](https://mlsanity.com/paper/2602.15184/) (2026)
- [WLNO: Wavelet-Laplace Neural Operator for Solving Partial Differential Equations](https://mlsanity.com/paper/2605.24658/) (2026)

## 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.