Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations

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.

💡 Research Summary

This paper introduces the Physics‑Informed Laplace Neural Operator (PILNO), a neural operator framework that combines an enhanced Laplace Neural Operator backbone with physics‑informed training to achieve high accuracy in small‑data regimes and robust out‑of‑distribution (OOD) generalization.
The authors first identify a limitation of the original Laplace Neural Operator (LNO): its pole‑residue parameterization simultaneously governs both transient and steady‑state responses, which can restrict expressivity in deep architectures. To overcome this, they propose the Advanced LNO (ALNO). ALNO retains the transient branch in its original pole‑residue form—preserving interpretable time‑scale parameters—while replacing the steady‑state branch with a Fourier‑style spectral multiplier, essentially an FNO layer. This decoupling yields a model that is both physically interpretable and significantly more expressive.
Building on ALNO, PILNO incorporates three key innovations for physics‑informed learning:

1. Physics‑based loss terms: PDE residuals, boundary‑condition (BC) residuals, and initial‑condition (IC) residuals are computed directly on the network’s predictions using automatic differentiation or consistent finite differences. These are combined with a supervised data loss when paired input‑output samples are available, weighted by hyper‑parameters λ_data, λ_pde, λ_bc, and λ_ic.
2. Virtual inputs: An unlabeled ensemble of input functions is generated to span a wide spectral range (e.g., varying correlation lengths and frequency content). For these virtual inputs only the physics residuals are enforced, providing abundant supervision without requiring ground‑truth solutions. This strategy explicitly expands the training distribution, improving data efficiency and OOD robustness.
3. Temporal‑causality weighting: A time‑decaying factor (e.g., w(t)=exp(−γt)) multiplies the physics loss, emphasizing early‑time dynamics during training. This mitigates error accumulation in long‑time predictions and stabilizes optimization for time‑dependent PDEs.
   The overall training objective is
    L(θ)=λ_data L_data+λ_pde L_pde+λ_bc L_bc+λ_ic L_ic,
   with the option to set λ_data=0 for a purely physics‑driven regime.
   The authors evaluate PILNO on four benchmark PDEs: (i) Burgers’ equation (initial‑condition‑to‑solution map), (ii) Darcy flow (coefficient‑to‑solution map), (iii) a reaction‑diffusion system (forcing‑to‑solution map), and (iv) a forced Korteweg‑de Vries (KdV) equation (combined forcing, BC, IC map). For each problem they conduct experiments with extremely limited training data (N_train ≤ 27) and test on both in‑distribution and OOD inputs whose spectral characteristics differ from the training set.
   Key findings include:

• Accuracy: PILNO consistently reduces the mean L2 error by 30–45 % compared with a purely data‑driven LNO across all benchmarks, especially when virtual inputs are used.
• OOD performance: In OOD tests, error growth is roughly halved relative to baselines, demonstrating that the virtual‑input strategy successfully teaches the operator to respect the governing physics beyond the training distribution.
• Stability and variance: Temporal‑causality weighting leads to smoother loss curves and lower error accumulation in long‑time forecasts. Moreover, the standard deviation of errors across multiple random seeds is markedly reduced, indicating improved reproducibility.
• Ablation studies: Removing either the virtual‑input set or the temporal‑causality weighting degrades performance, confirming that both components contribute complementary benefits.
  The paper’s contributions are threefold: (1) a novel ALNO architecture that decouples transient pole‑residue dynamics from a flexible Fourier steady‑state branch, (2) a physics‑informed training regime augmented with label‑free virtual inputs for data‑efficient learning, and (3) a time‑decay weighting scheme that prioritizes early dynamics and stabilizes training for time‑dependent problems.
  Limitations noted by the authors include the need to manually design the spectral range of virtual inputs, sensitivity of the decay hyper‑parameter, and the fact that all experiments are conducted on synthetic data. Future work is suggested in automated virtual‑input generation, extension to multi‑physics and multi‑scale problems, and validation on real experimental datasets. Overall, PILNO represents a significant step toward neural operators that are both data‑efficient and robust to distribution shifts, opening avenues for reliable surrogate modeling in scientific and engineering applications.