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.

翻译：我们提出了拉普拉斯神经算子（Laplace Neural Operator, LNO），该算子利用拉普拉斯变换对输入空间进行分解。与傅里叶神经算子（Fourier Neural Operator, FNO）不同，LNO能够处理非周期信号、捕捉瞬态响应，并呈现指数级收敛特性。LNO在输入与输出空间之间引入了极点-留数关系，从而提升了可解释性与泛化能力。本文中，我们证明了在逼近三种常微分方程（杜芬振荡器、受驱重力摆和洛伦兹系统）及三种偏微分方程（欧拉-伯努利梁、扩散方程和反应扩散系统）的解时，LNO中单个拉普拉斯层的逼近精度优于FNO中四个傅里叶模块的精度。值得注意的是，在无阻尼场景下，LNO在捕捉瞬态响应方面优于FNO。对于线性欧拉-伯努利梁和扩散方程，LNO对极点-留数公式的精确表示显著优于FNO。对于非线性反应扩散系统，LNO的误差小于FNO，这证明了将系统极点和留数作为网络参数用于算子学习的有效性。总体而言，我们的结果表明，LNO为学习在无限维空间之间映射函数的神经算子提供了一种有前景的新方法。