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[2602.12706] Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations
Summary
The paper presents the Physics-Informed Laplace Neural Operator (PILNO), a novel approach to solving partial differential equations (PDEs) that integrates governing physics into neural network training, enhancing performance in small-data scenarios.
Why It Matters
This research addresses the limitations of traditional data-driven models in solving PDEs, particularly in small-data regimes. By incorporating physics into the training process, PILNO improves accuracy and generalization, making it a significant advancement for applications in engineering and scientific computing.
Key Takeaways
- PILNO enhances the Laplace Neural Operator by embedding physics into training.
- The model improves accuracy in small-data settings and reduces variability.
- It effectively targets out-of-distribution input functions for better generalization.
- PILNO utilizes virtual inputs and temporal-causality weighting for robust training.
- The approach shows promise across various benchmarks, including Burgers' equation and Darcy flow.
Computer Science > Machine Learning arXiv:2602.12706 (cs) [Submitted on 13 Feb 2026] Title:Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations Authors:Heechang Kim, Qianying Cao, Hyomin Shin, Seungchul Lee, George Em Karniadakis, Minseok Choi View a PDF of the paper titled Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations, by Heechang Kim and 5 other authors View PDF HTML (experimental) 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 superv...