LNO: Laplace Neural Operator for Solving Differential Equations

TL;DR

The paper tackles the challenge of learning neural operators for time-dependent ODEs/PDEs with parameter variability, where traditional solvers are costly and Fourier-based operators struggle with non-periodic and transient signals. It proposes the Laplace neural operator (LNO), which parameterizes the kernel in the Laplace domain and uses a pole-residue formulation to model both transient and steady-state responses. Across six benchmarks (Duffing, forced gravity pendulum, Lorenz, Euler-Bernoulli beam, diffusion, reaction-diffusion), LNO consistently achieves higher accuracy than FNO and often outperforms GRU, especially in undamped or multi-pattern regimes. The work provides an interpretable, physically grounded neural operator with improved generalization, offering a promising direction for efficient surrogates mapping between infinite-dimensional function spaces.

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.

Figures (8)

• Figure 1: (a) Schematic representation of the full architecture of Laplace neural operator (LNO). We start from an input function $\mathbf f(t)$ and follow the following steps. $1.$ Lift the input function to a higher dimension by a shallow neural network $\mathcal{P}$. $2.$ Apply a Laplace layer and a local linear transform $W$. $3.$ Project the output, $u(t)$, back to the target dimension employing a shallow neural network, $\mathcal{Q}$. $(b)$ Laplace layer: start from input $V(s)$. Top row: apply the pole-residue method to compute the transient response residues $\gamma_n$ based on system poles $\mu_n$ and residues $\beta_n$; express the transient response in the Laplace domain. Bottom row: apply the pole-residue method to compute the steady-state response residues $\lambda_{\ell}$ based on input poles $i\omega_{\ell}$ and residues $\alpha_{\ell}$; express the steady-state response in Laplace domain.
• Figure 2: A schematic representation of the examples and the subsequent experiment scenarios under consideration in this work. Shown are representative plots of the input/output functions.
• Figure 3: Relative $\mathcal{L}_2$ error in the test cases for all the ODE and PDE cases and for different scenarios considered in each example. The plot shows the mean and the standard deviation of the error that has been computed based on five independent training trials.
• Figure 4: Pointwise error plots of responses for two representative test samples drawn from three ODE experiments. The ground truth is plotted by red curves and the pointwise error for LNO, FNO, and GRU are presented by blue curves.
• Figure 5: Duffing oscillator: Comparison of training and testing losses and responses obtained using LNO and FNO: $(a)$ learning curve of the system without damping, $(b)$ representative response obtained from the system without damping, for test cases, $(c)$ learning curve of the system with damping $c=0.5$, $(d)$ representative response obtained from the system with damping $c=0.5$, for test cases.