LNO: Laplace Neural Operator for Solving Differential Equations

TL;DR

The paper introduces the Laplace neural operator (LNO), which uses Laplace transforms to improve the approximation of differential equations, especially for non-periodic signals and transient responses, outperforming Fourier neural operators.

Contribution

The paper presents the novel LNO architecture that leverages Laplace transforms and pole-residue relationships, offering better accuracy and interpretability over Fourier neural operators.

Findings

LNO outperforms FNO in approximating solutions of ODEs and PDEs.

LNO effectively captures transient responses in undamped systems.

LNO's pole-residue formulation yields significantly better results for linear systems.

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.