Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations

TL;DR

PILNO tackles data efficiency and OOD generalization in neural-operator surrogates for time-dependent PDEs by augmenting an Advanced Laplace Neural Operator backbone with physics-informed losses, a virtual-input ensemble, and temporal-causality weighting. The method decomposes dynamics into transient and steady-state components via a pole-residue representation and a Fourier multiplier, improving expressivity while preserving interpretability. Empirical results across Burgers’ equation, Darcy flow, reaction–diffusion systems, and forced KdV show PILNO achieves substantially lower errors in small-data regimes, stronger OOD generalization, and reduced run-to-run variability, with virtual inputs and TCW driving notable gains in data-scarce and time-dependent settings. These findings suggest physics-informed operator learning can provide robust, data-efficient surrogates for parametric PDEs, with potential impact on rapid prototyping and uncertainty quantification in engineering and physics.

Abstract

Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs). However, purely data-driven models often require extensive training data and can generalize poorly, especially in small-data regimes and under unseen (out-of-distribution) input functions that are not represented in the training data. To address these limitations, we propose the Physics-Informed Laplace Neural Operator (PILNO), which enhances the Laplace Neural Operator (LNO) by embedding governing physics into training through PDE, boundary condition, and initial condition residuals. To improve expressivity, we first introduce an Advanced LNO (ALNO) backbone that retains a pole-residue transient representation while replacing the steady-state branch with an FNO-style Fourier multiplier. To make physics-informed training both data-efficient and robust, PILNO further leverages (i) virtual inputs: an unlabeled ensemble of input functions spanning a broad spectral range that provides abundant physics-only supervision and explicitly targets out-of-distribution (OOD) regimes; and (ii) temporal-causality weighting: a time-decaying reweighting of the physics residual that prioritizes early-time dynamics and stabilizes optimization for time-dependent PDEs. Across four representative benchmarks -- Burgers' equation, Darcy flow, a reaction-diffusion system, and a forced KdV equation -- PILNO consistently improves accuracy in small-data settings (e.g., N_train <= 27), reduces run-to-run variability across random seeds, and achieves stronger OOD generalization than purely data-driven baselines.

Figures (19)

• Figure 1: Schematic of PILNO architecture. Given input data $a$, a lifting operator $\mathcal{P}$ embeds the input into a higher-dimensional latent space; the temporal module applies an advanced laplace layer, and a projection map $\mathcal{Q}$ returns the output field. Training uses supervised data (when available) together with physics-based losses: PDE residuals (via derivatives such as $u_t, u_x, u_{xx}$) and boundary/initial-condition penalties. The total objective combines data loss with PDE residuals to enforce the governing equations during training.
• Figure 2: Relative $L_2$ errors over the number of training samples for Burgers' equation. Points denote means and shaded bands indicate $\pm1$ standard deviation over five random seeds. As $N_\text{train}$ decreases, the error of LNO increases sharply ($9.35\%$ at $N_\text{train}=25$), whereas PILNO maintains lower error level ($1.42\%$ at $N_\text{train}=25$).
• Figure 3: Worst–case time–slicing comparisons on the test set for the two LNO cases with the largest relative $L_2$ error. PILNO approximates the reference solution more accurately than LNO.
• Figure 4: Generalization across initial–condition length–scales. Heatmaps show relative $L^2$ error when training at length–scale $\ell_{\text{train}}$ (vertical axis) and testing at $\ell_{\text{test}}$ (horizontal axis), for (a) LNO and (b) PILNO. PILNO maintains low errors even when extrapolating from smooth ($\ell=5.0$) to oscillatory ($\ell=0.5$) regimes. PILNO consistently demonstrates better generalization, showing robust predictive performance even on data distributions not included during training.
• Figure 5: Comparison at $N_\text{train}=10$ (Darcy flow) on $61\times61$ resolution. LNO attains a relative $L_2$ error of 14.67%, whereas PILNO achieves 1.23%, indicating substantially better performance by PILNO under extremely limited training data.